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Acceleration

Acceleration

In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. It is thus a vector quantity with dimension length/time². In SI units, this is meter/second².

Explanation

To accelerate an object is to change its velocity over a period of time. In this strict scientific sense, acceleration can have positive and negative values – respectively called acceleration and deceleration (or retardation) in common speech – as well as change of direction. Acceleration is defined technically as "the rate of change of velocity of an object with respect to time" and is given by the equation :

\mathbf =

where :a is the acceleration vector :v is the velocity vector expressed in m/s :t is time expressed in seconds. This equation gives a the units of m/(s·s), or m/s² (read as "metres per second per second", or "metres per second squared"). An alternative equation is: :

\mathbf =

where :ā is the average acceleration (m/s²) :u is the initial velocity (m/s) :v is the final velocity (m/s) :t is the time interval (s) Transverse acceleration (perpendicular to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have :

\mathbf = - \frac \frac = - \omega^2 \mathbf

One common unit of acceleration is g, one g being the acceleration caused by the gravity of Earth at sea level at 45° latitude (Paris), or about 9.81 m/s². Jerk is the rate of change of an object's acceleration over time. In classical mechanics, acceleration

a \

is related to force

F \

and mass

m \

(assumed to be constant) by way of Newton's second law: :

F = m \cdot a

As a result of its invariance under the Galilean transformations, acceleration is an absolute quantity in classical mechanics.

Relation to relativity

After defining his theory of special relativity, Albert Einstein realized that forces felt by objects undergoing constant acceleration are indistinguishable from those in a gravitational field, and thus defined general relativity that also explained how gravity's effects could be limited by the speed of light. If you accelerate away from your friend, you could say (given your frame of reference) that it is your friend who is accelerating away from you, although only you feel any force. This is also the basis for the popular Twin paradox, which asks why only one twin ages when moving away from his sibling at near light-speed and then returning, since the aging twin can say that it is the other twin that was moving. General relativity solved the "why does only one object feel accelerated?" problem which had plagued philosophers and scientists since Newton's time (and caused Newton to endorse absolute space). In special relativity, only inertial frames of reference (non-accelerated frames) can be used and are equivalent; general relativity considers all frames, even accelerated ones, to be equivalent. With changing velocity, accelerated objects exist in warped space (as do those that reside in a gravitational field). Therefore, frames of reference must include a description of their local spacetime curvature to qualify as complete. Acceleration can be measured using an accelerometer.

References

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External links and references

- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- [http://www.sandia.gov/LabNews/labs-accomplish/2005/pulp.html Experiments on Z produced a world-record peak velocity of 34 km/s] (that is about 76,000 mph)
- [http://www.sandia.gov/media/NewsRel/NR2001/flyer.htm Magnetic field shocklessly shoots pellets 20 times faster than rifle bullet] Category:Physical quantity Category:Classical mechanics ko:가속도 ja:加速度 simple:Acceleration th:ความเร่ง

Derivative

In mathematics, the derivative is one of the two central concepts of calculus. (The other is the integral; the two are related via the fundamental theorem of calculus.) The simplest type of derivative is the derivative of a real-valued function of a single real variable. It has several interpretations:
- The derivative gives the slope of a tangent to the graph of the function at a point. In this way, derivatives can be used to determine many geometrical properties of the graph, such as concavity or convexity.
- The derivative provides a mathematical formulation of rate of change; it measures the rate at which the function's value changes as the function's argument changes. This derivative is the kind usually encountered in a first course on calculus, and historically was the first to be discovered. However, there are also many generalizations of the derivative. The remainder of this article discusses only the simplest case (real-valued functions of real numbers).

Differentiation and differentiability

In physical terms, differentiation expresses the rate at which a quantity, y, changes with respect to the change in another quantity, x, on which it has a functional relationship. Using the symbol Δ to refer to change in a quantity, this rate is defined as a limit of difference quotients :

\frac

as Δx approaches 0. In Leibniz's notation for derivatives, the derivative of y with respect to x is written :

\frac

suggesting the ratio of two infinitesimal quantities. The above expression is pronounced in various ways such as "dy by dx" or "dy over dx". The form "dy dx" is also used conversationally, although it may be confused with the notation for element of area. Modern mathematicians do not bother with "dependent quantities", but simply state that differentiation is a mathematical operation on functions. The precise definition of this operation (which therefore need not deal with infinitesimal quantities) is given as: :

\lim_\frac.

A function is differentiable at a point x if its derivative exists at that point; a function is differentiable on an interval if it is differentiable at every x within the interval. If a function is not continuous at x, then there is no tangent line and the function is therefore not differentiable at x; however, even if a function is continuous at x, it may not be differentiable there. In other words, differentiability implies continuity, but not vice versa. One famous example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function. The derivative of a differentiable function can itself be differentiable. The derivative of a derivative is called a second derivative. Similarly, the derivative of a second derivative is a third derivative, and so on.

Newton's difference quotient

The derivative of a function f at x is geometrically the slope of the tangent line to the graph of f at x. Without the concept which we are about to define, it is impossible to directly find the slope of the tangent line to a given function, because we only know one point on the tangent line, namely (x, f(x)). Instead, we will approximate the tangent line with multiple secant lines that have progressively shorter distances between the two intersecting points. When we take the limit of the slopes of the nearby secant lines in this progression, we will get the slope of the tangent line. The derivative is then defined by taking the limit of the slope of secant lines as they approach the tangent line. tangent tangent To find the slopes of the nearby secant lines, choose a small number h. h represents a small change in x, and it can be either positive or negative. The slope of the line through the points (x,f(x)) and (x+h,f(x+h)) is :

.

This expression is Newton's difference quotient. The derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line: :

f'(x)=\lim_.

difference quotient If the derivative of f exists at every point x in the domain, we can define the derivative of f to be the function whose value at a point x is the derivative of f at x. Since immediately substituting 0 for h results in division by zero, calculating the derivative directly can be unintuitive. One technique is to simplify the numerator so that the h in the denominator can be cancelled. This happens easily for polynomials; see calculus with polynomials. For almost all functions however, the result is a mess. Fortunately, many guidelines exist.

Notations for differentiation

Lagrange's notation

The simplest notation for differentiation that is in current use is due to Joseph Louis Lagrange and uses the prime mark:

Leibniz's notation

The other common notation is Leibniz's notation for differentiation which is named after Leibniz. For the function whose value at x is the derivative of f at x, we write: : $\frac.$ We can write the derivative of f at the point a in two different ways: : $\frac\left.\right|_ = \left(\frac\right)(a).$ If the output of f(x) is another variable, for example, if y=f(x), we can write the derivative as: : $\frac.$ Higher derivatives are expressed as : $\frac$ or $\frac$ for the n-th derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is: : $\frac$ which we can loosely write as: : $\left(\frac\right)^3 \left(f(x)\right) =
\frac \left(f(x)\right).$ Dropping brackets gives the notation above. Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember, because the "du" terms appear symbolically to cancel: : $\frac = \frac \cdot \frac.$ (In the popular formulation of calculus in terms of limits, the "du" terms cannot literally cancel, because on their own they are undefined; they are only defined when used together to express a derivative. In nonstandard analysis, however, they can be viewed as infinitesimal numbers that cancel.)
Newton's notation
Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the function name: : $\dot = \frac = x'(t)$ : $\ddot = x$ (t) and so on. Newton's notation is mainly used in mechanics, normally for time derivatives such as velocity and acceleration, and in ODE theory. It is usually only used for first and second derivatives.

Euler's notation

Euler's notation uses a differential operator, denoted as D, which is prefixed to the function with the variable as a subscript of the operator: This notation can also be abbreviated when taking derivatives of expressions that contain a single variable. The subscript to the operator is dropped and is assumed to be the only variable present in the expression. In the following examples, u represents any expression of a single variable: Euler's notation is useful for stating and solving linear differential equations.

Critical points

Points on the graph of a function where the derivative is undefined or equals zero are called critical points or sometimes stationary points (in the case where the derivative equals zero). If the second derivative is positive at a critical point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum. Taking derivatives and solving for critical points is often a simple way to find local minima or maxima, which can be useful in optimization. In fact, local minima and maxima can only occur at critical points. This is related to the extreme value theorem.

Physics

Arguably the most important application of calculus to physics is the concept of the "time derivative"—the rate of change over time—which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:
- Velocity (instantaneous velocity; the concept of average velocity predates calculus) is the derivative (with respect to time) of an object's position.
- Acceleration is the derivative (with respect to time) of an object's velocity.
- Jerk is the derivative (with respect to time) of an object's acceleration. For example, if an object's position

p(t) = -16t^2 + 16t + 32

; then, the object's velocity is

\dot p(t) = p'(t) = -32t + 16

; the object's acceleration is

\ddot p(t) = p

(t) = -32; and the object's jerk is $p(t) = 0.$ If the velocity of a car is given, as a function of time, then, the derivative of said function with respect to time describes the acceleration of said car, as a function of time.
Algebraic manipulation
Messy limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via algebraic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient.
- Constant rule: The derivative of any constant is zero.
- Constant multiple rule: If c is some real number; then, the derivative of $cf(x)$ equals c multiplied by the derivative of f(x) (a consequence of linearity below).
- Linearity: $(af + bg)' = af' + bg'$ for all functions f and g and all real numbers a and b.
- Power rule: If $f(x) = x^r$ , for some real number r; $f'(x) = rx^$ .
- Product rule: $(fg)' = f'g + fg'$ for all functions f and g.
- Quotient rule: $(f/g)' = (f'g - fg')/(g^2)$ unless g is zero.
- Chain rule: If $f(x) = h(g(x))$ , then $f'(x) = h'(g(x)) g'(x)$ .
- Inverse function: If $g(x) = f^(x)$ , and $f(x)$ is injective, then $g'(x) = 1/f'(f^(x))$ .
- Derivative of one variable with respect to another when both are functions of a third variable: Let $x = f(t)$ and $y = g(t)$ . Now $d y/d x = (d y/d t)/(d x/d t)$ . This is the chain rule in the Leibniz notation.
- Implicit differentiation: If $f(x,y) = 0$ is an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y). In addition, the derivatives of some common functions are useful to know. See the table of derivatives. As an example, the derivative of : $f(x) = 2x^4 + \sin (x^2) - \ln (x)\;e^x + 7$ is : $f'(x) = 8x^3 + 2x\cos (x^2) - \frac\;e^x - \ln (x)\;e^x.$
Using derivatives to graph functions
Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all critical points are local extrema; for example, f(x)=x³ has a critical point at x=0, but it has neither a maximum nor a minimum there. The first derivative test and the second derivative test provide ways to determine if the critical points are maxima, minima or neither. In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension at local extrema. In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a saddle point, and if none of these cases hold then the test is inconclusive (e.g., eigenvalues of 0 and 3). Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side.
Generalizations
Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Some people pronounce the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'. The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles. In order to differentiate all continuous functions and much more, one defines the concept of distribution. For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument. For example, the function f(x + iy) = x + 2iy satisfies the latter, but not the first. See also the article on holomorphic functions.
See also

- Derivative (examples)
- Derivative (generalizations)
- Partial derivative
- Total derivative
- Table of derivatives
- Smooth function
- Differintegral
- Automatic differentiation
External links

- [http://wims.unice.fr/wims/wims.cgi?module=tool/analysis/function.en WIMS Function Calculator] makes online calculation of derivatives; this software enables also interactive exercises.
References

- Spivak, Michael; Calculus (3rd edition, 1994) Publish or Perish Press. ISBN 0914098896. Explains why all this works.
- Thompson, Silvanus Phillips, Calculus made easy : being a very-simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the differential calculus and the integral calculus New York : St. Martin's Press, 1998 ISBN 0312185480. Introduced by Martin Gardner. "What one fool can do, another can."
- Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (2003). Calculus of a Single Variable: Early Transcendental Functions (3rd edition). Houghton Mifflin Company. ISBN 061822307X.
- Anton, Howard (1980). Calculus with analytical geometry.. New York:John Wiley and Sons. ISBN 0-471-03248-4.
- ko:미분 ja:微分 simple:Derivative th:อนุพันธ์

Vector (spatial)

:This article discusses vectors that have a particular relation to the spatial coordinates. For a generalization, see vector space. In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). Although it is often described by a number of "components", each of which is dependent upon the particular coordinate system being used, a vector is an object with properties which do not depend on the coordinate system used to describe it. A common example of a vector is force — it has a magnitude and an orientation in three dimensions (or however many spatial dimensions one has), and multiple forces sum according to the parallelogram law. A spatial vector can be formally defined by its relationship to the spatial coordinate system under rotations. Alternatively, it can be defined in a coordinate-free fashion via a tangent space of a three-dimensional manifold in the language of differential geometry. These definitions are discussed in more detail below. A spatial vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry). Vectors are the building blocks of vector fields and vector calculus. The word vector is also now used for more general concepts (see also vector and generalizations below), but this article describes the original spatial meaning except where otherwise noted.

Definitions

Informally, a vector is a quantity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction, often represented graphically by an arrow. Examples are "moving north at 90 km/h" or "pulling towards the center of Earth with a force of 70 newtons". The notion of having a "magnitude" and "direction" is formalized by saying that the vector has components that transform like the coordinates under rotations. That is, if the coordinate system undergoes a rotation described by a rotation matrix R, so that a coordinate vector x is transformed to x' = Rx, then any other vector v is similarly transformed via v' = Rv. This ensures the invariance of the operations dot product, Euclidean norm, cross product, gradient, divergence, curl, and scalar triple product, and trivially for vector addition and subtraction, and scalar multiplication. The terms scalar and vector as used here include pseudoscalars and pseudovectors or axial vectors (see also below). Accordingly, let, for example, each of two vectors be expressed as three space coordinates, and apply the formula for the cross product, resulting in three coordinates, which represent a third vector. If we rewrite the two vectors in rotated coordinates, and apply the formula for the cross product again, then the result is the original cross product in terms of rotated coordinates. Also, let, for example, a vector field be expressed as three space coordinate functions of three variables, and apply the formula for the curl based on these functions, resulting in three additional functions, which represent a second vector field. If we rewrite the original vector field in terms of rotated position coordinates and correspondingly rotated coordinates for the vector function values, and apply the formula for the curl based on these functions, then the result is the rewritten version of the original curl: also in terms of rotated position coordinates and correspondingly rotated coordinates for the vector function values. The same applies for dot product, gradient, divergence, vector addition and scalar multiplication. For these, also reflection in a plane can be applied. The scalars involved should not be transformed (e.g. in the case of a rotation by 180°, the scalar should not be multiplied by -1). Thus even in 1D we have to distinguish scalars and vectors: 2 × 3 = 6 can be interpreted as a scalar multiplication or a dot product, but not as a product of two vectors. Similarly differentiation in 1D can be interpreted as a gradient or a divergence: one of the two functions is scalar and one a vector, and the argument is a vector, ensuring invariance under inversion of the vectors without changing the scalars. Since rotation of the three Cartesian coordinate axes changes the formulas the same as an inverse rotation of the field itself, we can also conclude:
- if the same rotation is applied to two vectors, then the cross product is correspondingly rotated, but the dot product remains the same
- rotation of a scalar field results in a correspondingly rotated vector field for the gradient
- rotation of a vector field results in a correspondingly rotated scalar field for the divergence and a correspondingly rotated vector field for the curl where rotation of a scalar field involves only rotation of the position vectors, while rotation of a vector field involves also a corresponding rotation of the vector field values. Note that the concept of corresponding rotations applies even if different coordinate systems are used for field values and position vectors, so that e.g. for one we multiply by an orthogonal matrix and for the other we add an angle to an angle coordinate. In order to use the usual formulas, e.g. to compute mechanical work, the x-axis of forces should be in the same direction as the x-axis of position, etc. When, as described above, coordinate rotations of position are accompanied by corresponding coordinate rotations of forces, this property is preserved. On the other hand, the origin of forces is simply at the zero force (no force), while the origin of position can be chosen as desired. For example, work depends on displacement, which is the difference of positions and therefore does not depend on the origin. Position and function value of a vector field are often, but not necessarily, expressed in similar coordinate systems. For example gravitational field strength due to a particular point mass may be

g_r = -9.8 (r/r_0)^ m/s^2

, with both the function value and the position vector in spherical coordinates. For the position vector the origin is chosen here at the center of the point mass; for the field strength the origin is simply at "zero field strength" anyway. How the other two coordinates are chosen does not matter in this case, because the field does not depend on them, and the field has no components in their directions. More generally, a vector is a tensor of contravariant rank one. In differential geometry, the term vector usually refers to quantities that are closely related to tangent spaces of a differentiable manifold (assumed to be three-dimensional and equipped with a positive definite Riemannian metric). (A four-vector is a related concept when dealing with a 4 dimensional spacetime manifold in relativity.) Examples of vectors include displacement, velocity, electric field, momentum, force, and acceleration. Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations). The magnitude of any vector is a scalar. A related concept is that of a pseudovector (or axial vector). This is a quantity that transforms like a vector under proper rotations, but gains an additional sign flip under improper rotations. Examples of pseudovectors include magnetic field, torque, and angular momentum. (This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying symmetry properties.) To distinguish from pseudo/axial vectors, an ordinary vector is sometimes called a polar vector. See also parity (physics). Sometimes, one speaks informally of bound or fixed vectors, which are vectors additionally characterized by a "base point". Most often, this term is used for position vectors (relative to an origin point). More generally, however, the physical interpretation of a particular vector can be parameterized by any number of quantities.

Examples in one dimension

A force may be "15N to the right", with coordinate 15N if the basis vector is to the right, and −15N if the basis vector is to the left. The magnitude of the vector is 15N in both cases. A displacement may be "4m to the right", with coordinate 4m if the basis vector is to the right, and −4m if the basis vector is to the left. The magnitude of the vector is 4m in both cases. The work done by the force in the case of this displacement is 60J in both cases. The force and displacement are vectors, the magnitudes are scalars, and the coordinates are neither.

Generalizations

In mathematics, a vector is any element of a vector space over some field. The spatial vectors of this article are a very special case of this general definition (they are not simply any element of R^d in d dimensions), which includes a variety of mathematical objects (algebras, the set of all functions from a given domain to a given linear range, and linear transformations). Note that under this definition, a tensor is a special vector!

Representation of a vector

Symbols standing for vectors are usually printed in boldface as a; this is also the convention adopted in this encyclopedia. Other conventions include

\vec

or a, especially in handwriting. Alternately, some use a tilde (~) placed under the vector. The length or magnitude or norm of the vector a is denoted by |a|. Vectors are usually shown in graphs or other diagrams as arrows, as illustrated below: Image:vecab.png Here the point A is called the tail, base, start, or origin; point B is called the head, tip, endpoint, or destination. The length of the arrow represents the vector's magnitude, while the direction in which the arrow points represents the vector's direction. If a vector is itself spatial, the length of the arrow depends on a dimensionless scale. If it represents e.g. a force, the "scale" is of physical dimension length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2cm, the scales are 1:250 and 1m:50N respectively. Equal length of vectors of different dimension has no particular significance unless there is some proportionality constant inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance. In the figure above, the arrow can also be written as

\overrightarrow

or AB In order to calculate with vectors, the graphical representation is too cumbersome. Vectors in a n-dimensional Euclidean space can be represented as a linear combination of n mutually perpendicular unit vectors. In this article, we will consider R³ as an example. In R³, we usually denote the unit vectors parallel to the x-, y- and z-axes by i, j and k respectively. Any vector a in R³ can be written as a = a₁i + a₂j + a₃k with real numbers a₁, a₂ and a₃ which are uniquely determined by a. Sometimes a is then also written as a 3-by-1 or 1-by-3 matrix: :

= \begin
 a_1\\
 a_2\\
 a_3\\
\end

= \begin
 a_1 & a_2 & a_3 \\
\end

even though this notation suppresses the dependence of the coordinates a₁, a₂ and a₃ on the specific choice of coordinate system i, j and k.

Length of a vector

The length of the vector a = a₁i + a₂j + a₃k can be computed with the Euclidian norm :

\left\|\mathbf\right\|=\sqrt

which is a consequence of the Pythagorean theorem.

Vector equality

Two vectors are said to be equal if they have the same magnitude and direction. However if we are talking about bound vector, then two bound vectors are equal if they have the same base point and end point. For example, the vector i + 2j + 3k with base point (1,0,0) and the vector i+2j+3k with base point (0,1,0) are different bound vectors, but the same (unbounded) vector.

Vector addition and subtraction

Let a=a₁i + a₂j + a₃k and b=b₁i + b₂j + b₃k. The sum of a and b is: :

\mathbf+\mathbf
=(a_1+b_1)\mathbf
+(a_2+b_2)\mathbf
+(a_3+b_3)\mathbf

The addition may be represented graphically by placing the start of the arrow b at the tip of the arrow a, and then drawing an arrow from the start of a to the tip of b. The new arrow drawn represents the vector a + b, as illustrated below: Pythagorean theorem This addition method is sometimes called the parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are bound vectors, then the addition is only defined if a and b have the same base point, which will then also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c). The difference of a and b is: :

\mathbf-\mathbf
=(a_1-b_1)\mathbf
+(a_2-b_2)\mathbf
+(a_3-b_3)\mathbf

Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the ends of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. That arrow represents the vector a − b, as illustrated below: parallelogram If a and b are bound vectors, then the subtraction is only defined if they share the same base point which will then also become the base point of their difference. This operation deserves the name "subtraction" because (a − b) + b = a. In physics, vectors of different physical dimension may occur in the same diagram. However, adding or subtracting them (graphically or otherwise) is meaningless.

Scalar multiplication

A vector may also be multiplied by a real number r. In mathematics numbers are often called scalars to distinguish them from vectors, and this operation is therefore called scalar multiplication. The resulting vector is: :

r\mathbf=(ra_1)\mathbf
+(ra_2)\mathbf
+(ra_3)\mathbf

The length of ra is |r||a|. If the scalar is negative, it also changes the direction of the vector by 180^o. Two examples (r = -1 and r = 2) are given below: real number Here it is important to check that the scalar multiplication is compatible with vector addition in the following sense: r(a + b) = ra + rb for all vectors a and b and all scalars r. One can also show that a - b = a + (-1)b. The set of all geometrical vectors, together with the operations of vector addition and scalar multiplication, satisfies all the axioms of a vector space. Similarly, the set of all bound vectors with a common base point forms a vector space. This is where the term "vector space" originated. In physics, scalars also have a unit. The scale of acceleration in the diagram is e.g. 2 m/s² : cm, and that of force 5 N : cm. Thus a scale ratio of 2.5 kg : 1 is used for mass. Similarly, if displacement has a scale of 1:1000 and velocity of 0.2 cm : 1 m/s, or equivalently, 2 ms : 1, a scale ratio of 0.5 : s is used for time.

Unit vector

Main article: Unit vector A unit vector is any vector with a length of one. If you have a vector of arbitrary length, you can use it to create a unit vector. This is known as normalizing a vector. Unit vector To normalize a vector a = [a₁, a₂, a₃], scale the vector by the inverse of its length ||a||. That is: :

\mathbf=\frac=\frac\mathbf+\frac\mathbf+\frac\mathbf

Dot product

Main article: Dot product The dot product of two vectors a and b (sometimes called inner product, or, since its result is a scalar, the scalar product) is denoted by a·b and is defined as: :

\mathbf\cdot\mathbf
=\left\|\mathbf\right\|\left\|\mathbf\right\|\cos(\theta)

where ||a|| and ||b|| denote the norm (or length) of a and b, and θ is the measure of the angle between a and b (see trigonometric function for an explanation of cosine). Geometrically, this means that a and b are drawn with a common start point and then the length of a is multiplied with the length of that component of b that points in the same direction as a. This operation is often useful in physics; for instance, work is the dot product of force and displacement.

Cross product

The cross product (also vector product or outer product) differs from the dot product primarily in that the result of a cross product of two vectors is a vector. While everything that was said above can be generalized in a straightforward manner to more than three dimensions, the cross product is only meaningful in three dimensions (although a related product exists in seven dimensions - see below). The cross product, denoted a×b, is a vector perpendicular to both a and b and is defined as: :

\mathbf\times\mathbf
=\left\|\mathbf\right\|\left\|\mathbf\right\|\sin(\theta)\,\mathbf

where θ is the measure of the angle between a and b, and n is a unit vector perpendicular to both a and b. The problem with this definition is that there are two unit vectors perpendicular to both b and a. Which vector is the correct one depends upon the orientation of the vector space, i.e. on the handedness of the coordinate system. The coordinate system i, j, k is called right handed, if the three vectors are situated like the thumb, index finger and middle finger (pointing straight up from your palm) of your right hand. Graphically the cross product can be represented by this figure Image:crossproduct.png In such a system, a×b is defined so that a, b and a×b also becomes a right handed system. If i, j, k is left-handed, then a, b and a×b is defined to be left-handed. Because the cross product depends on the choice of coordinate systems, its result is referred to as a pseudovector. Fortunately, in nature cross products tend to come in pairs, so that the "handedness" of the coordinate system is undone by a second cross product. The length of a×b can be interpreted as the area of the parallelogram having a and b as sides.

Scalar triple product

The scalar triple product (also called the box product or mixed triple product) isn't really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is denoted by (a b c) and defined as: :

(\mathbf\ \mathbf\ \mathbf)
=\mathbf\cdot(\mathbf\times\mathbf)

It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors a, b and c are oriented like the coordinate system i, j and k. In coordinates, if the three vectors are thought of as rows, the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows. The scalar triple product is linear in all three entries and anti-symmetric in the following sense: : Technically, the scalar triple product is not a scalar, it is a pseudoscalar: under a coordinate inversion (x goes to −x), it flips sign.

Vectors as directional derivatives

A vector may also be defined as a directional derivative: consider a function

f(x^\alpha)

and a curve

x^\alpha (\sigma)

. Then the directional derivative of

f

is a scalar defined as

\frac = \frac\frac.

where the index

\alpha

is summed over the appropriate number of dimensions (e.g. from 1 to 3 in 3-dimensional Euclidian space, from 0 to 3 in 4-dimensional spacetime, etc.). Then consider a vector tangent to

x^\alpha (\sigma)

t^\alpha = \frac.

We can rewrite the directional derivative in differential form (without a given function

f

) as

\frac = t^\alpha\frac.

Therefore any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative. We can therefore define a vector precisely: :

\mathbf \equiv a^\alpha \frac.

External links

- [http://wwwppd.nrl.navy.mil/nrlformulary/vector_identities.pdf Online vector identities] (PDF)
- Vectors at Wikibooks Category:Abstract algebra Category:Vector calculus Category:Linear algebra Category:Introductory physics ko:벡터 ja:ベクトル (数学)

Time

Attempting to understand Time has long been a prime occupation for philosophers, scientists and artists. There are widely divergent views about its meaning, hence it is difficult to provide an uncontroversial and clear definition of time. The Oxford English Dictionary defines it as "the indefinite continued progress of existence and events in the past, present, and future, regarded as a whole". Another standard dictionary definition is "a non-spatial linear continuum wherein events occur in an apparently irreversible order." This article looks at some of the main philosophical and scientific issues relating to time. The measurement of time has also occupied scientists and technologists, and was a prime motivation in astronomy. Time is also a matter of significant social importance, having economic value ("time is money") as well as personal value due to an awareness of the limited time in each day and in our lives. Units of time have been agreed upon to quantify the duration of events and the intervals between them. Regularly recurring events and objects with apparently periodic motion have long served as standards for units of time - such as the apparent motion of the sun across the sky, the phases of the moon, the swing of a pendulum.

Philosophy of time

Main article: Philosophy of space and time; Ontology In ancient thought, Zeno's paradoxes challenged the conception of infinite divisibility, and eventually led to the development of calculus. Parmenides (of whom Zeno was a follower) believed that time, motion, and change were illusions, basing this on a rather interesting argument. More recently, McTaggart held a similar belief. Newton believed time and space form a container for events, which is as real as the objects it contains. In contrast, Leibniz believed that time and space are a conceptual apparatus describing the interrelations between events. Leibniz and others thought of time as a fundamental part of an abstract conceptual framework, together with space and number, within which we sequence events, quantify their duration, and compare the motions of objects. In this view, time does not refer to any kind of entity that "flows", that objects "move through", or that is a "container" for events. The bucket argument proved problematic for Leibniz, and his account fell into disfavour, at least amongst scientists, until the development of Mach's principle. Modern physics views the curvature of spacetime around an object as much a feature of that object as are its mass and volume. Immanuel Kant, in the Critique of Pure Reason, described time as an a priori notion that allows us (together with other a priori notions such as space) to comprehend sense experience. With Kant, neither space nor time are conceived as substances, but rather both are elements of a systematic framework necessarily structuring the experiences of any rational agent. Spatial measurements are used to quantify how far apart objects are, and temporal measurements are used to quantify how far apart events occur. Nietzsche, inspired by the concept of eternal return in his book Thus Spoke Zarathustra, argued that time possesses a circular characteristic. Postulating an infinite past, "all things" must have come to pass therein; the same for an infinite future. In Existentialism, time is considered fundamental to the question of being, in particular by the philosopher Martin Heidegger.

Contemporary theses in the philosophy of time

In contempoary philosophy there has been a very active debate over the nature of time, especially in light of the big changes in physics since the 1920s. Contributors include Ned Markosian, Ted Sider, Quentin Smith, and L. Nathan Oaklander. Two major theses have been developed, along with some hybrids. There is no real consensus among philosophers about which, if any, is correct. The two major theories can be summed up as follows: 1. A-theory of time: Presentism: Oaklander writes: "[A] version of the pure A-theory, known as "", purports to avoid… the problem of change... According to presentism, only the present exists. Thus, it is not the case that, say, O is green and [then] O is red [if, for example, O is a tomato]." (Oaklander, L. Nathan. In Smith, Quentin, and Oaklander, L. Nathan. 1995. Time, Change, and Freedom. New York: Routledge. 2004, 27.) 2. B-theory of time: Eternalism: the following passage from L. Nathan Oaklander sums this up

…[T]ime [involves] events strung out along a series united to one another by the relations of earlier than, later and simultaneity… The events in the temporal series are fixed in that they never change their position relative to each other… It has become customary to call the entire series of events spread out along the time-line from earlier to later, the “B-series.” When viewed solely in terms of the B-series, time is thought of as static or unchanging for there is nothing about temporal relations between events that changes... Time not only has a static aspect, it also has a transitory aspect. In addition to conceiving of time in terms of events standing in temporal relations, we also conceive of time and the events in time as moving or passing from the far future to the near future, from the hear future to the present, and then from present they recede into the more and more distant past… When events are ordered in terms of the notions of past, present, or future they form what is called an “A-series.” It should be noted, of course, that the A- and B-series are not really “two” different series of events, but the same series ordered in two different ways. (Oaklander 2004,Page 69)

Time in physics

never change Main article: Time in physics Time is currently one of the few fundamental quantities (quantities which cannot be defined via other quantities because there is nothing more fundamental known at present). Thus, similar to definition of other fundamental quantities (like space and mass), time is defined via measurement. Currently, the standard time interval (called conventional second, or simply second) is defined as 9 192 631 770 oscillations of a hyperfine transition in the ¹³³Cs atom. Prior to Albert Einstein's relativistic physics, time and space had been treated as distinct dimensions; Einstein linked time and space into spacetime. Einstein showed that people traveling at different speeds will measure different times for events and different distances between objects, though these differences are minute unless one is traveling at a speed close to that of light. Many subatomic particles exist for only a fixed fraction of a second in a lab relatively at rest, but some that travel close to the speed of light can be measured to travel further and survive longer than expected. According to the special theory of relativity, in the high-speed particle's frame of reference, it exists for the same amount of time as usual, and the distance it travels in that time is what would be expected for that velocity. Relative to a frame of reference at rest, time seems to "slow down" for the particle. Relative to the high-speed particle, distances seems to shorten. Even in Newtonian terms time may be considered the fourth dimension of motion; but Einstein showed how both temporal and spatial dimensions can be altered (or "warped") by high-speed motion. Einstein (The Meaning Of Relativity - 1968): "Two events taking place at the points A and B of a system K are simultaneous if they appear at the same instant when observed from the middle point, M, of the interval AB. Time is then defined as the ensemble of the indications of similar clocks, at rest relatively to K, which register the same simultaneously."

Measurement

Present day standards

The standard unit for time is the SI second, from which larger units are defined like the minute, hour, and day. Because they do not use the decimal system, and because of the occasional need for a leap-second, the minute, hour, and day are "non-SI" units, but are officially accepted for use with the International System. There are no fixed ratios between seconds (or days) on the one hand and months and years on the other hand -- months and years having significant variations in length. Despite its great social importance, the week is not mentioned even as a "non-SI" unit. ([http://www1.bipm.org/utils/en/pdf/si-brochure.pdf See external pdf file: The International System of Units].) The measurement of time is so critical to the functioning of our modern societies that it is coordinated at an international level. The basis for scientific time is a continuous count of seconds based on atomic clocks around the world, known as International Atomic Time (TAI). This is the yardstick for other time scales including Coordinated Universal Time (UTC) which is the basis for civil time. The 60 base used for seconds, minutes and hours is all the remains of the ancient Phoenician counting base, using 60 as the equivalent of 10, or 100 in modern times. A 60 base is known as sexagesimal.

Chronology

Another form of time measurement consists of studying the past. Events in the past can be ordered in a sequence (creating a chronology), and be put into chronological groups (periodization). One of the most important systems of periodization is Geologic time, which is a system of periodizing the events that shaped the Earth and its life. Chronology, periodization, and interpretation of the past are together known as the study of history.

Psychology

Different people may judge identical lengths of time quite differently. Time can "fly"; that is, a long period of time can seem to go by very quickly. Likewise, time can seem to "drag," as in when one performs a boring task. The psychologist Jean Piaget called this form of time perception "lived time". Time appears to go fast when sleeping, or, to put it differently, time seems not to have passed while asleep. Time also appears to pass more quickly as one gets older. For example, a day for a child seems to last longer than a day for an adult. One possible reason for this is that with increasing age, each segment of time is an increasingly smaller percentage of the person's total experience. Altered states of consciousness are sometimes characterised by a different estimation of time. Some psychoactive substances--such as entheogens--may also dramatically alter a person's temporal judgement. In explaining his theory of relativity, Albert Einstein is often quoted as saying that although sitting next to a pretty girl for an hour feels like a minute, placing one's hand on a hot stove for a minute feels like an hour. This is intended to introduce the listener to the concept of the interval between two events being perceived differently by different observers.

Use of time

The use of time is an important issue in understanding human behaviour, education, and travel behaviour. The question concerns how time is allocated across a number of activities (such as time spent at home, at work, shopping, etc.). Time use changes with technology, as the television or the Internet created new opportunities to use time in different ways. However, some aspects of time use are relatively stable over long periods of time, such as the amount of time spent traveling to work, which despite major changes in transport, has been observed to be about 20-30 minutes one-way for a large number of cities over a long period of time. This has led to the disputed time budget hypothesis. Time management is the organization of tasks or events by first estimating how much time a task will take to be completed, when it must be completed, and then adjusting events that would interfere with its completion so that completion is reached in the appropriate amount of time. Calendars and day planners are common examples of time management tools. Arlie Russell Hochschild and Norbert Elias have written on the use of time from a sociological perspective.

References

- Oxford English Dictionary - [http://www.askoxford.com/concise_oed/time?view=uk]

External links

Perception of time

- [http://plato.stanford.edu/entries/time-experience/ The Experience and Perception of Time]
- [http://cogprints.ecs.soton.ac.uk/archive/00003125/ Subjective Perception of Time and a Progressive Present Moment: The Neurobiological Key to Unlocking Consciousness]
- [http://www.primitivism.com/time.htm Time and Its Discontents]
- [http://www.ericdigests.org/2003-5/time.htm Time and Learning]
- [http://mixingmemory.blogspot.com/2004/12/by-request-time-perception-i.html Time Perception I] and [http://mixingmemory.blogspot.com/2004/12/time-perception-ii-cognitive-factors.html II]
- [http://theorderoftime.org/ The Order of Time: Platform for an Alternative Time Consciousness]
- [http://www.chabad.org/article.asp?AID=74335 What is Time?] An elucidation of the Lubavitcher Rebbe's comments on the topic.

Physics

- [http://physics.nist.gov/GenInt/Time/world.html A walk through Time]
- [http://pages.britishlibrary.net/lobster/tmx Time Travel and Multi-Dimensionality]
- [http://arxiv.org/abs/physics/0310055 Time and classical and quantum mechanics: Indeterminacy vs. discontinuity]
- [http://www.sankey.ws/time.html Time as a universal consequence of quanta]

Timekeeping

- [http://tycho.usno.navy.mil/systime.html Different systems of measuring time]
- [http://physics.nist.gov/cuu/Units/outside.html non-SI units]
- [http://www1.bipm.org/en/scientific/tai/time_server.html UTC/TAI Timeserver]
- [http://tycho.usno.navy.mil/leapsec.html Leapsecond]
- [http://www.intuitor.com/hex/hexclock.html Hex Time]
- [http://www.florencetime.net Florencetime.net]
- [http://news.bbc.co.uk/2/hi/science/nature/3486160.stm BBC article on shortest time ever measured]
- [http://www.awi-net.org American Watchmakers-Clockmakers Institute]
- [http://www.timeanddate.com/worldclock/ The World Clock - Time Zones]

Miscellaneous

- [http://www.boost.org/doc/html/date_time.html Boost Date-Time Library -- Powerful C++ Library for date-time manipulation]
- [http://www.cyclesresearchinstitute.org/ Cycles Research Institute]
- [http://www.timeticker.com/ TimeTicker and the time tickers...]
- [http://www.welt-zeit-uhr.de/worldtime.php World Time and Zones]
- [http://www.timetools.co.uk Time Servers] NTP Time Servers provide accurate timing for computers and computer networks.

Meter per second squared

Metres per second squared is the SI derived unit of acceleration, defined by distance or displacement in metres divided by time in seconds and again divided by time in seconds. It can be either a scalar or a vector, depending on whether it is derived from distance or displacement. The unit is written in symbols as m/s² or m·s^-2 or m s^-2. It may be better understood when phrased as "metres per second, per second", i.e. the increase in speed (in metres per second), that is achieved each second. The SI derived unit of force, the newton, is equal to one "kilogram metre per second squared", meaning that a force of one newton is needed to give an acceleration of one metre per second squared to a mass of one kilogram. This is an illustration of Newton's second law (see Newton's laws of motion) in its simplest form, where acceleration is defined as the rate of change of velocity. Category:Units of acceleration Category:SI derived units ja:メートル毎秒毎秒

Transverse

The term transverse means "side-to-side", as opposed to longitudinal, which means "front-to-back".
- In automotive engineering, the term transverse refers to an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle. See transverse engine.
- transverse wave See also: velocity, acceleration.

Perpendicular

:For the Deep Purple album, see Purpendicular. right Perpendicular is a geometric term that may be used as a noun or adjective. The fundamental meaning pertains to the position of straight lines relative to one another. Two lines are said to be perpendicular if they meet at a right angle. Note that two line segments positioned at 90° to one another are perpendicular only if they meet. Two lines are considered perpendicular if the product of their slopes is -1. Naturally, if a line is given, then a perpendicular is any line at a ninety-degree angle to that line. This is an important property in geometry and trigonometry since important properties accrue to line systems containing right angles. When graphing, the convention is to use either an X and Y axis, or to use an X, Y, and Z axis, which are defined as being mutually perpendicular. Right triangles, too, include two perpendicular lines and so have special properties, which are the foundation of trigonometry. Compare to parallel.

Formula

When given 2 straight lines A and B, with A: y = ωx + a; B: y = ω'x + b A and B are perpendicular in an orthonormal base (where X-axis and Y-axis are perpendicular and the distance between (0,0) and (1,0) is equal to the distance between (0,0) and (0,1)) if ω
- ω'=-1. This fact can also lead to funny (but correct) results with imaginary lines: e.g. the line y = ix is perpendicular with a line with ω = i, we see now that the line y = ix is perpendicular with itself! This seems odd but is nevertheless correct. For people interested in this strange fact and who like to give this a little thought, it needs to be said that imaginary lines can be seen as cubes in a 4-dimension space.

Centripetal acceleration

Centripetal force is the force accelerating an object toward the center of a circular path as the object goes around the circle. An object can travel in a circle only if there is a centripetal force on it. In the case of an orbiting satellite the centripetal force is its weight and acts toward the satellite's primary; in the case of an object at the end of a rope, the centripetal force is the tension of the rope and acts towards whatever the rope is anchored to. In the case of a spinning object, internal tensile stress provides the centripetal force that keeps the object together. Centripetal force must not be confused with centrifugal force. In an inertial reference frame (not rotating or accelerating), the centripetal force accelerates a particle in such a way that it moves along a circular path. In a corotating reference frame, a particle in circular motion has zero velocity. In this case, the centripetal force appears to be exactly cancelled by a pseudo-force, the centrifugal force. Centripetal forces are true forces, appearing in inertial reference frames; centrifugal forces appear only in rotating frames. Centripetal force must not be confused with central force either. Objects moving in a straight line with constant speed also have constant velocity. However, an object moving in an arc with constant speed has a changing direction of motion. As velocity is a vector of speed and direction, a changing direction implies a changing velocity. The rate of this change in velocity is the centripetal acceleration. Differentiating the velocity vector gives the direction of this acceleration towards the center of the circle. In the plane of motion we have: :

\mathbf = - \frac \hat = - \frac \frac = - \omega^2 \mathbf

By Newton's second law of motion, as there is an acceleration there has to be a force in the direction of the acceleration. This is the centripetal force, and is in the plane of motion equal to: :

\mathbf = - \frac \hat = - \frac \frac = - m \omega^2 \mathbf

(where m is mass, v is velocity, r is radius of the circle, and the minus sign denotes that the vector points to the center of the circle and ω = v / r is the angular velocity) In 3D notation we can write: :

\boldsymbol F_c = m \boldsymbol\omega \times   (\boldsymbol\omega \times   \boldsymbol r )

, where

\boldsymbol\omega

is the vector angular velocity of the rotation and

\boldsymbol r

is a vector from an arbitrary point on the rotation axis to the body (with mass

m

). Centripetal means towards the center.

Proof of law

Simply use a polar coordinate system, assume a constant radius, and take two derivatives. Let r(t) be a vector that describes the position of a point mass as a function of time. Since we are assuming uniform circular motion, let r(t) = R·u_r, where R is a constant (the radius of the circle) and u_r is the unit vector pointing from the origin to the point mass. In terms of Cartesian unit vectors: :

u_r = cos(\theta)u_x + sin(\theta)u_y \,

Note: unlike in cartesian coordinates where the unit vectors are constants, in polar coordinates the direction of the unit vectors depend on the angle between the x_axis and the point being described; the angle θ. So we take the first derivative to find velocity: :

v = R \frac  \,

v = R \frac u_\theta \,

v = R \omega u_\theta \,

where ω is the angular velocity (just a short way of writing dθ/dt), u_θ is the unit vector that is perpendicular to u_r that points in the direction of increasing θ. In cartesian terms: u_θ = -sin(θ) u_x + cos(θ) u_y This result for the velocity is good because it matches our expectation that the velocity should be directed around the circle, and that the magnitude of the velocity should be ωR. Taking another derivative, we find that the acceleration, a is: :

a = R \left( \frac  u_\theta - \omega^2 u_r \right) \,

since the motion is uniform, the magnitude of v is constant, and thus there can be no dv/dt that points in the same direction as v. This fact implies ω is constant which simplifies the equation to: :

a = -R \omega^2 u_r \,

A simple substitution brings us to the equation above.

References

-
- Category:Force Category:Mechanics ko:구심력 ja:回転運動

Gravity

Gravity is the force of attraction between massive particles. Weight is determined by the mass of an object and its location in a gravitational field. While a great deal is known about the properties of gravity, the ultimate cause of the gravitational force remains an open question. General relativity is the most successful theory of gravitation to date. It postulates that mass and energy curve space-time, resulting in the phenomenon known as gravity. The effect of the bending of spacetime is often misunderstood as most people seem to prefer to think of a falling object as accelerating when the facts do not support that assumption. Skydivers do not feel any acceleration (other than from wind resistance). Gravity is acceleration.

F=\dot=m\dot+\dotv\,\!

means (if the mass is unvarying) that there must be a force that causes a mass to accelerate. For a rocket ship, that is the rocket engine. For the earth, it is the compression of the mass between something on the surface of the earth and the earth's center of mass. The acceleration is in relation to spacetime in that the weight one feels is one's resistance to deviating from one's path in spacetime. The same holds true in the rocket ship except that a rocket engine supplies the force to accelerate an occupant from his spacetime path. There is no difference between the weight he feels because of gravity or the rocket.

Newton's law of universal gravitation

Newton's law of universal gravitation states the following: :Every object in the Universe attracts every other object with a force directed along the line of centers of mass for the two objects. This force is proportional to the product of their masses and inversely proportional to the square of the separation between the centers of mass of the two objects. Given that the force is along the line through the two masses, the law can be stated symbolically as follows. :

F = - G \frac

where: :F is the magnitude of the (repulsive) gravitational force between two objects :G is the gravitational constant, that is approximately : G = 6.67 × 10⁻¹¹ N m² kg^-2 :m₁ is the mass of first object :m₂ is the mass of second object :r is the distance between the objects It can be seen that this repulsive force F is always negative, and this means that the net attractive force is positive. The minus sign is used to hold the same value meaning as in the Coulomb's Law, where a positive force as result means repulsion between two charges. Thus gravity is proportional to the mass of each object, but has an inverse square relationship with the distance between the centres of each mass. Strictly speaking, this law applies only to point-like objects. If the objects have spatial extent, the force has to be calculated by integrating the force (in vector form, see below) over the extents of the two bodies. It can be shown that for an object with a spherically-symmetric distribution of mass, the integral gives the same gravitational attraction on masses outside it as if the object were a point mass.¹ This law of universal gravitation was originally formulated by Isaac Newton in his work, the Principia Mathematica (1687). Professor William Whewell of Cambridge University, author of History of the Inductive Sciences (1837) stated: ::The law of gravitation is indisputably and incomparably the greatest scientific discovery ever made, whether we look at the advance which it involved, the extent of the truth disclosed, or the fundamental and satisfactory nature of this truth. [In A Treasury of Science ed. Harlow Shapley et al, Harper & Bros. NY: 1946] The history of gravitation as a physical concept is considered in more detail below.

Vector form

below Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude. In this formula, quantities in bold represent vectors. :

\mathbf_ =
 G 
 \, \mathbf_

\mathbf_ =
 - G 
 \, \mathbf_

where :F₁₂ is the force on object 1 due to object 2 :G is the gravitational constant :m₁ and m₂ are the masses of the objects 1 and 2 :r₂₁ = | r₂ − r₁ | is the distance between objects 2 and 1 :

\mathbf_ \equiv \frac

is the unit vector from object 2 to 1 It can be seen, that the vector form of the equation is the same as the scalar form, except for the vector value of F and the unit vector. Also, it can be seen that F₁₂ = − F₂₁. Gravitational acceleration is given by the same formula except for one of the factors m: :

\mathbf =
  G 
  \, \mathbf

Gravitational field

The gravitational field is a vector field that describes the gravitational force an object of given mass experiences in any given place in space. It is a generalization of the vector form, which becomes particularly useful if more than 2 objects are involved (such as a rocket between the Earth and the Moon). For 2 objects (e.g. object 1 is a rocket, object 2 the Earth), we simply write

\mathbf r

instead of

\mathbf r_

and

m

instead of

m_1

and define the gravitational field

\mathbf g(\mathbf r)

as: :

\mathbf g(\mathbf r) =
 G 
 \, \mathbf

so that we can write: :

\mathbf( \mathbf r) = m \mathbf g(\mathbf r)

This formulation is independent of the objects causing the field. The field has units of force divided by mass; in SI, this is N·kg⁻¹.

Problems with Newton's theory

Although Newton's formulation of gravitation is quite accurate for most practical purposes, it has a few problems:

Theoretical concerns

- There is no prospect of identifying the mediator of gravity. Newton himself felt the inexplicable action at a distance to be unsatisfactory (see "Newton's reservations" below).
- Newton's theory requires that gravitational force is transmitted instantaneously. Given classical assumptions of the nature of space and time, this is necessary to preserve the conservation of angular momentum observed by Johannes Kepler. However, it is in direct conflict with Einstein's theory of special relativity which places an upper limit—the speed of light in vacuum—on the velocity at which signals can be transmitted.

Disagreement with observation

- Newton's theory does not fully explain the precession of the perihelion of the orbit of the planet Mercury. There is a 43 arcsecond per century discrepancy between the Newtonian prediction (resulting from the gravitational tugs of the other planets) and the observed precession.
- The predicted deflection of light by gravity is only half as much as observations of this deflection, which were made after General Relativity was developed in 1915.
- The observed fact that gravitational and inertial masses are the same for all bodies is unexplained within Newton's system. General relativity takes this as a postulate. See equivalence principle.

Newton's reservations

It's important to understand that while Newton was able to formulate his law of gravity in his monumental work, he was deeply uncomfortable with the notion of "action at a distance" which his equations implied. He never, in his words, "assigned the cause of this power". In all other cases, he used the phenomenon of motion to explain the origin of various forces acting on bodies, but in the case of gravity, he was unable to experimentally identify the motion that produces the force of gravity. Moreover, he refused to even offer a hypothesis as to the cause of this force on grounds that to do so was contrary to sound science. He lamented the fact that "philosophers have hitherto attempted the search of nature in vain" for the source of the gravitational force, as he was convinced "by many reasons" that there were "causes hitherto unknown" that were fundamental to all the "phenomena of nature". These fundamental phenomena are still under investigation and, though hypotheses abound, the definitive answer is yet to be found. While it is true that Einstein's hypotheses are successful in explaining the effects of gravitational forces more precisely than Newton's in certain cases, he too never assigned the cause of this power, in his theories. It is said that in Einstein's equations, "matter tells space how to curve, and space tells matter how to move", but this new idea, completely foreign to the world of Newton, does not enable Einstein to assign the "cause of this power" to curve space any more than the Law of Universal Gravitation enabled Newton to assign its cause. In Newton's own words: :I wish we could derive the rest of the phenomena of nature by the same kind of reasoning from mechanical principles; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto attempted the search of nature in vain. If science is eventually able to discover the cause of the gravitational force, Newton's wish could eventually be fulfilled as well. It should be noted that here, the word "cause" is not being used in the same sense as "cause and effect" or "the defendant caused the victim to die". Rather, when Newton uses the word "cause," he (apparently) is referring to an "explanation". In other words, a phrase like "Newtonian gravity is the cause of planetary motion" means simply that Newtonian gravity explains the motion of the planets. See Causality and Causality (physics).

Einstein's theory of gravitation

Einstein's theory of gravitation answered the problems with Newton's theory noted above. In a revolutionary move, his theory of general relativity (1915) stated that the presence of mass, energy, and momentum causes spacetime to become curved. Because of this curvature, the paths that objects in inertial motion follow can "deviate" or change direction over time. This deviation appears to us as an acceleration towards massive objects, which Newton characterized as being gravity. In general relativity however, this acceleration or free fall is actually inertial motion. So objects in a gravitational field appear to fall at the same rate due to their being in inertial motion while the observer is the one being accelerated. (This identification of free fall and inertia is known as the Equivalence principle.) The relationship between the presence of mass/energy/momentum and the curvature of spacetime is given by the Einstein field equations. The actual shapes of spacetime are described by solutions of the Einstein field equations. In particular, the Schwarzschild solution (1916) describes the gravitational field around a spherically symmetric massive object. The geodesics of the Schwarzschild solution describe the observed behavior of objects being acted on gravitationally, including the anomalous perihelion precession of Mercury and the bending of light as it passes the Sun. Arthur Eddington found observational evidence for the bending of light passing the Sun as predicted by general relativity in 1919. Subsequent observations have confirmed Eddington's results, and observations of a pulsar which is occulted by the Sun every year have permitted this confirmation to be done to a high degree of accuracy. There have also in the years since 1919 been numerous other tests of general relativity, all of which have confirmed Einstein's theory.

Units of measurement and variations in gravity

tests of general relativity. (ESA image)]] Gravitational phenomena are measured in various units, depending on the purpose. The gravitational constant is measured in newtons times metre squared per kilogram squared. Gravitational acceleration, and acceleration in general, is measured in metres per second squared or in non-SI units such as galileos, gees, or feet per second squared. The acceleration due to gravity at the Earth's surface is approximately 9.81 m/s², more precise values depending on the location. A standard value of the Earth's gravitational acceleration has been adopted, called g_n. When the typical range of interesting values is from zero to tens of metres per second squared, as in aircraft, acceleration is often stated in multiples of g_n. When used as a measurement unit, the standard acceleration is often called "gee", as g can be mistaken for g, the gram symbol. For other purposes, measurements in millimetres or micrometres per second squared (mm/s² or µm/s²) or in multiples of milligals or milligalileos (1 mGal = 1/1000 Gal), a non-SI unit still common in some fields such as geophysics. A related unit is the eotvos, which is a cgs unit of the gravitational gradient. Mountains and other geological features cause subtle variations in the Earth's gravitational field; the magnitude of the variation per unit distance is measured in inverse seconds squared or in eotvoses. Typical variations with time are 2 µm/s² (0.2 mGal) during a day, due to the tides, i.e. the gravity due to the Moon and the Sun. A larger variation in the effect of gravity occurs when we move from the equator to the poles. The effective force of gravity decreases as the distance from the equator decreases, due to the rotation of the Earth, and the resulting centrifugal force and flattening of the Earth. The centrifugal force causes an effective force 'up' which effectively counteracts gravity, while the flattening of the Earth causes the poles to be closer to the center of mass of the Earth. It is also related to the fact that the Earth's density changes from the surface of the planet to its centre. The sea-level gravitational acceleration is 9.780 m/s² at the equator and 9.832 m/s² at the poles, so an object will exert about 0.5% more force due to gravity at sea level at the poles than at sea level at the equator [http://curious.astro.cornell.edu/question.php?number=310].

Comparison with electromagnetic force

The gravitational interaction of protons is approximately a factor 10³⁶ weaker than the electromagnetic repulsion. This factor is independent of distance, because both interactions are inversely proportional to the square of the distance. Therefore on an atomic scale mutual gravity is negligible. However, the main interaction between common objects and the Earth and between celestial bodies is gravity, because at this scale matter is electrically neutral: even if in both bodies there were a surplus or deficit of only one electron for every 10¹⁸ protons and neutrons this would already be enough to cancel gravity (or in the case of a surplus in one and a deficit in the other: double the interaction). However, the main interactions between the charged particles in cosmic plasma (that makes up over 99% of the universe by volume), are electromagnetic forces. In terms of Planck units: the charge of a proton is 0.085, while the mass is only . From that point of view, the gravitational force is not small as such, but because masses are small. The relative weakness of gravity can be demonstrated with a small magnet picking up pieces of iron. The small magnet is able to overwhelm the gravitational interaction of the entire Earth. Similarly, when doing a chin-up, the electromagnetic interaction within your muscle cells is able to overcome the force induced by Earth on your entire body. Gravity is small unless at least one of the two bodies is large or one body is very dense and the other is close by, but the small gravitational interaction exerted by bodies of ordinary size can fairly easily be detected through experiments such as the Cavendish torsion bar experiment. Cavendish torsion bar experiment Further reading
- Jefimenko, Oleg D., "Causality, electromagnetic induction, and gravitation : a different approach to the theory of electromagnetic and gravitational fields". Star City [West Virginia] : Electret Scientific Co., c1992. ISBN 0917406095
- Heaviside, Oliver, "[http://www.as.wvu.edu/coll03/phys/www/Heavisid.htm A gravitational and electromagnetic analogy]". The Electrician, 1893.

Gravity and quantum mechanics

It is strongly believed that three of the four fundamental forces (the strong nuclear force, the weak nuclear force, and the electromagnetic force) are manifestations of a single, more fundamental force. Combining gravity with these forces of quantum mechanics to create a theory of quantum gravity is currently an important topic of research amongst physicists. General relativity is essentially a geometric theory of gravity. Quantum mechanics relies on interactions between particles, but general relativity requires no exchange of particles in its explanation of gravity. Scientists have theorized about the graviton (a messenger particle that transmits the force of gravity) for years, but have been frustrated in their attempts to find a consistent quantum theory for it. Many believe that string theory holds a great deal of promise to unify general relativity and quantum mechanics, but this promise has yet to be realized. It is notable that in general relativity gravitational radiation (which under the rules of quantum mechanics must be composed of gravitons) is only created in situations where the curvature of spacetime is oscillating, such as for co-orbiting objects. The amount of gravitational radiation emitted by the solar system and its planetary systems is far too small to measure. However, gravitational radiation has been indirectly observed as an energy loss over time in binary pulsar systems such as PSR1913+16). It is believed that neutron star mergers and black hole formation may create detectable amounts of gravitational radiation. Gravitational radiation observatories such as LIGO have been created to study the problem. No confirmed detections have been made of this hypothetical radiation, but as the science behind LIGO is refined and as the instruments themselves are endowed with greater sensitivity over the next decade, this may change.

Experimental tests of theories

Today General Relativity is accepted as the standard description of gravitational phenomena. (Alternative theories of gravitation exist but are more complicated than General Relativity.) General Relativity is consistent with all currently available measurements of large-scale phenomena. For weak gravitational fields and bodies moving at slow speeds at small distances, Einstein's General Relativity gives almost exactly the same predictions as Newton's law of gravitation. Crucial experiments that justified the adoption of General Relativity over Newtonian gravity were the classical tests: the gravitational redshift, the deflection of light rays by the Sun, and the precession of the orbit of Mercury. More recent experimental confirmations of General Relativity were the (indirect) deduction of gravitational waves being emitted from orbiting binary stars, the existence of neutron stars and black holes, gravitational lensing, and the convergence of measurements in observational cosmology to an approximately flat model of the observable Universe, with a matter density parameter of approximately 30% of the critical density and a cosmological constant of approximately 70% of the critical density. The equivalence principle, the postulate of general relativity that presumes that inertial mass and gravitational mass are the same, is also under test. Past, present, and future tests are discussed in the equivalence principle section. Even to this day, scientists try to challenge General Relativity with more and more precise direct experiments. The goal of these tests is to shed light on the yet unknown relationship between Gravity and Quantum Mechanics. Space probes are used to either make very sensitive measurements over large distances, or to bring the instruments into an environment that is much more controlled than it could be on Earth. For example, in 2004 a dedicated satellite for gravity experiments, called Gravity Probe B, was launched to test general relativity's predicted frame-dragging effect, among others. Also, land-based experiments like LIGO and a host of "bar detectors" are trying to detect gravitational waves directly. A space-based hunt for gravitational waves, LISA, is in its early stages. It should be sensitive to low frequency gravitational waves from many sources, perhaps including the Big Bang.

Acceleration

Explanation

Relation to relativity

References

External links and references

Derivative

Differentiation and differentiability

Newton's difference quotient

Notations for differentiation

Lagrange's notation

Leibniz's notation

Newton's notation

Euler's notation

Critical points

Physics

Algebraic manipulation

Using derivatives to graph functions

Generalizations

See also

External links

References

Vector (spatial)

Definitions

Examples in one dimension

Generalizations

Representation of a vector

Length of a vector

Vector equality

Vector addition and subtraction

Scalar multiplication

Unit vector

Dot product

Cross product

Scalar triple product

Vectors as directional derivatives

See also

External links

Time

Philosophy of time

Contemporary theses in the philosophy of time

Time in physics

Measurement

Present day standards

Chronology

Psychology

Use of time

See also

General units of time

Special units of time

Time measurement and horology

Theory and study of time

References

External links

Perception of time

Physics

Timekeeping

Miscellaneous

Further reading

Meter per second squared

Transverse

Perpendicular

Formula

See also

Centripetal acceleration

Proof of law

See also

References

Gravity

Newton's law of universal gravitation

Vector form

Gravitational field

Problems with Newton's theory

Theoretical concerns

Disagreement with observation

Newton's reservations

Einstein's theory of gravitation

Units of measurement and variations in gravity

Comparison with electromagnetic force

Gravity and quantum mechanics

Experimental tests of theories