Frame of reference

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A frame of reference is a particular perspective from which the universe is observed. Specifically, in physics, it refers to a provided set of axes from which an observer can measure the position and motion of all points in a system, as well as the orientation of objects in it.

There are two types of reference frames: inertial and non-inertial. An inertial frame of reference is defined as one in which Newton's first law holds true. That is, one in which a free particle travels in a straight line (or more generally a geodesic) at constant speed. In three dimensional Euclidean space, using Cartesian co-ordinates, this means that

\cfrac{d^2x}{dt^2}=0 \qquad \cfrac{d^2y}{dt^2}=0 \qquad \cfrac{d^2z}{dt^2}=0

A non-inertial frame of reference, therefore, is one in which a free particle does not travel in a straight line at constant speed. For example a co-ordinate system centred at a point on the earth's surface. This frame of reference rotates around the centre of the earth which produces a fictitious force known as the coriolis force.

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[edit] Overview

Frames of reference are extremely important in the realm of physics for describing all types of phenomena. Choosing an appropriate reference frame and co-ordinate system may simplify the solution to a problem enormously. Let us consider for a second a situation which is relatively common in everyday life. Two cars are travelling along a road, both moving at a constant velocity. At exactly 2pm, they are separated by 200 metres. The car in front is travelling at 22 metres per second and the car behind is travelling at 30 metres per second. If we want to find out how long it will take the second car to catch up with the first, there are three obvious "frames of reference" that we could choose.

Firstly, we could observe the two cars from the side of the road. We define our "frame of reference" as follows. We stand on the side of the road right next to the second car at 2pm. We start a stop-clock at the exact moment that the second car passes us. Since neither of the cars are accelerating, we can determine their positions by the following formulae where x1(t) is the position of car one after time "t" and x2(t) is the position of car two after time "t".

x_1(t)=30t \quad x_2(t)=22t + 200

We want to find the time at which x1 = x2. Therefore we set x1 = x2 and solve for t. i.e.

30t=22t+200 \quad
8t = 200 \quad
t = 25 \quad seconds

Alternatively, we could choose a frame of reference situated in the first car. In this case, the first car is stationary and the second car is approaching from behind at a speed of 8 metres per second. In order to catch up to the first car, it will take a time of 200 \div 8 seconds. i.e. 25 seconds as before. Note how much easier the problem becomes by choosing a suitable frame of reference. It would have been possible to choose a rotating, accelerating frame of reference, moving in a complicated manner, but this would have served to complicate the problem unnecessarily. It is also necessary to note that one is able to convert measurements made in one co-ordinate system to another. For example, suppose that your watch is running five minutes fast compared to the local standard time. If you know that this is the case, when somebody asks you what time it is, you are able to deduct five minutes from the time displayed on your watch in order to obtain the correct time. The measurements that an observer makes about a system depend therefore on the observer's frame of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three).

It is important to note that there were a number of assumptions made about the various inertial frames of reference. Newton for instance believed in a concept known as universal time. This is best explained by an example. Suppose that you own two clocks, which both tick at exactly the same rate. You synchronise them so that they both display the exact same time. The two clocks are now separated and one clock is on a fast moving train, travelling at constant velocity towards the other. According to Newton, these two clocks will still tick at the same rate and will both show the same time. Newton says that the rate of time as measured in one frame of reference should be the same as the rate of time in another. That is, there exists a "universal" time and all other times in all other frames of reference will run at the same rate as this universal time irrespective of their position and velocity. This concept was later disproven by Einstein in his special theory of relativity (1905) where he developed transformations between inertial frames of reference based of their relative displacement and relative velocity (Lorentz transformations).

It is also important to note that the definition of inertial reference frame (defined as one in which a free particle travels in a straight line at constant speed) does not include a requirement that the inertial reference frame must exist in three dimensional Euclidean space. This was another of Newton's assumptions which would later be disproven. As an example of why this is important, let us consider the Non-Euclidean geometry of a sphere. In this geometry, two free particles may begin at the same point on the sphere, travelling with the same constant velocity in different directions. After a length of time, the two particles will collide at the opposite side of the sphere. Both free particles were travelling with a constant velocity and no forces were acting. No acceleration occurred and so Newton's first law held true. This means that the particles were in inertial frames of reference. Since no forces were acting, it was the geometry of the situation which caused the two particles to meet each other again. In a similar way, it is now believed that we exist in a four dimensional geometry known as spacetime. It is believed that the curvature of this 4D space is responsible for the way in which two bodies with mass will meet together even if no forces are acting. In Newtonian mechanics, this is explained by a force known as gravity.

[edit] Examples

For a simple example, consider two people standing, facing each other on either side of a North-South street. A car drives past them heading South. For the person facing East, the car was moving toward the right. However, for the person facing West, the car was moving toward the left. This discrepancy is due to the fact that the two people used two different frames of reference from which to investigate this system.

For a more complex example, consider Alfred, who is standing on the side of a road watching a car drive past him from left to right. In his frame of reference, Alfred defines the spot where he is standing as the origin, the road as the x-axis and the direction in front of him as the positive y-axis. To him, the car moves along the x axis with some velocity v in the positive x-direction. Alfred's frame of reference is considered an inertial frame of reference because he is not accelerating (ignoring effects such as Earth's rotation and gravity).

Now consider Betsy, the person driving the car. Betsy, in choosing her frame of reference, defines her location as the origin, the direction to her right as the positive x-axis, and the direction in front of her as the positive y-axis. In this frame of reference, it is Betsy who is stationary and the world around her that is moving - for instance, as she drives past Alfred, she observes him moving with velocity v in the negative y-direction. If she is driving north, then north is the positive y-direction; if she turns east, east becomes the positive y-direction.

Now assume Candace is driving her car in the opposite direction. As she passes by him, Alfred measures her acceleration and finds it to be a in the negative x-direction. Assuming Candace's acceleration is constant, what acceleration does Betsy measure? If Betsy's velocity v is constant, she is in an inertial frame of reference, and she will find the acceleration to be the same - in her frame of reference, a in the negative y-direction. However, if she is accelerating at rate A in the negative y-direction (in other words, slowing down), she will find Candace's acceleration to be a' = a - A in the negative y-direction - a smaller value than Alfred has measured. Similarly, if she is accelerating at rate A in the positive y-direction (speeding up), she will observe Candace's acceleration as a' = a + A in the negative y-direction - a larger value than Alfred's measurement.

Frames of reference are especially important in special relativity, because when a frame of reference is moving at some significant fraction of the speed of light, then the flow of time in that frame does not necessarily apply in another reference frame. The speed of light is considered to be the only true constant between moving frames of reference.

[edit] Nomenclature and notation

When working a problem involving one or more frames of reference it is common to designate an inertial frame of reference.

An accelerated frame of reference is often delineated as being the "primed" frame, and all variables that are dependent on that frame are notated with primes, e.g. x' , y' , a' .

The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as R. Given a point of interest that exists in both frames, the vector from the inertial origin to the point is called r, and the vector from the accelerated origin to the point is called r'. From the geometry of the situation, we get

\mathbf r = \mathbf R + \mathbf r'

Taking the first and second derivatives of this, we obtain

\mathbf v = \mathbf V + \mathbf v'
\mathbf a = \mathbf A + \mathbf a'

where V and A are the velocity and acceleration of the accelerated system with respect to the inertial system and v and a are the velocity and acceleration of the point of interest with respect to the inertial frame.

These equations allow transformations between the two coordinate systems; for example, we can now write Newton's second law as

\mathbf F = m\mathbf a = m\mathbf A + m\mathbf a'

When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decelerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or deceleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating frame of reference the manifestation of inertia appears to exert a force (either in centrifugal direction, or in tangential direction, the coriolis effect). In actual fact the force exerted on the object that keeps the object's motion in sync with the rotating frame elicits manifestation of inertia. If there is insufficient force to keep the object's motion in sync with the rotating frame, then seen from the perspective of the rotating frame there is an apparent acceleration. Whenever manifestation of inertia appears to act as a force it is labeled as a fictitious force. Inertia is very much real, of course, but unlike force it never accelerates an object. In General Relativity, fictitious forces due to acceleration are indistinguishable from gravity in the small (local region); even in the large, the two kinds of force can be distinguished only in special cases, such as static reference frames or reference frames asymptotic (at large distances) to Minkowskian, or at least static ones.

A common sort of accelerated reference frame is a frame that is both rotating and translating (an example is a frame of reference attached to a CD which is playing while the player is carried). This arrangement leads to the equation

\mathbf a = \mathbf a' + \dot{\boldsymbol\omega} \times \mathbf r' + 2\boldsymbol\omega \times \mathbf v' + \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') + \mathbf A_0

or, to solve for the acceleration in the accelerated frame,

\mathbf a' = \mathbf a - \dot{\boldsymbol\omega} \times \mathbf r' - 2\boldsymbol\omega \times \mathbf v' - \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') - \mathbf A_0

Multiplying through by the mass m gives

\mathbf F' = \mathbf F_\mathrm{physical} + \mathbf F'_\mathrm{transverse} + \mathbf F'_\mathrm{coriolis} + \mathbf F'_\mathrm{centripetal} - m\mathbf A_0

where

\mathbf F'_\mathrm{transverse} = -m\dot{\boldsymbol\omega} \times \mathbf r'
\mathbf F'_\mathrm{coriolis} = -2m\boldsymbol\omega \times \mathbf v' (Coriolis force)
\mathbf F'_\mathrm{centrifugal} = -m\boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r')=m(\omega^2 \mathbf r'- (\boldsymbol\omega \cdot \mathbf r')\boldsymbol\omega) (centrifugal force)

[edit] Particular frames of reference in common use

[edit] In fields other than Physics

[edit] See also

[edit] Footnote 1

Distortions can vary from place to place, with gravity appearing to be the common cause. In fact, General relativity predicts a frame-dragging effect (aka Lense-Thirring effect).

Retrieved from "http://en.wikipedia.org/wiki/Frame_of_reference"
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