Frame of Reference

A frame of reference is a set of three mutually perpendicular lines (directions or axes) used to specify the position of a particle or an event occurring in space.

The term ‘frame of reference’ is a synonym of an ‘observer’. With each observer, we can associate a frame of reference.

A frame of reference is a theoretical construct associated with an observer, which is at rest, relative to the observer, but is not the observer itself. Any phenomenon in the universe can be described relative to any frame of reference.

For, example, a box on a railway platform has zero velocity and zero acceleration relative to a person standing on the platform.

Suppose a train \( A \) is crossing the platform with a constant velocity \( \vec{v} \).

To an observer on such a train, the box appears to move with a constant velocity \( -\vec{v} \) and hence zero acceleration.

Suppose another train \( B \) goes past the platform not with uniform velocity, but with some acceleration \( \vec{a} \). For an observer on this train, the box has an acceleration \( -\vec{a} \).

We find that different statements about the same object (box in this case) arise from different states of motion of the observer/frame of reference.

For all the three frames of reference, the forces acting on the box are the same i.e. weight \( (W) \) of the box acting vertically downwards and normal reaction \( (R) \) acting vertically upwards. The two forces, being equal and opposite, add up to a zero net force.

For the platform’s frame of reference (which is at rest) and the frame of reference of train \( A \) (moving uniformly), zero net force corresponds to zero acceleration i.e. Newton’s first law is valid. But for the frame of reference of train \( B \) (moving with some acceleration), zero net force does not imply zero acceleration i.e. Newton’s first law is not valid.

A frame of reference, relative to which Newton’s first law is valid, is called an Inertial frame.

A frame of reference, relative to which Newton’s first law is not valid, is called a Non-inertial frame. It may also be considered as a frame of reference having accelerated motion is called a non-inertial frame of reference. In an accelerated frame of reference, an object experiences a fictitious force. For example, a bus moving along a circular track is an accelerated frame of reference. It is our experience that passengers sitting in such a bus are always thrown away from the center.

This force which tends to throw the passengers away from the center is called centrifugal force and is a fictitious force. The motion of objects in accelerated frames of reference cannot be studied by simply Newton’s laws of motion.

From the above discussion, we find that if a frame is inertial (like the platform’s frame), any frame of reference moving with constant velocity relative to it is also inertial. However, if a frame is accelerated relative to an inertial frame, it is bound to be non-inertial. Note that different inertial frames of reference are completely equivalent. It means that the laws of mechanics are identical for all of them.

For the sake of convenience, we use a Cartesian system of coordinates as a frame of reference. Usually, the point \( O \) is taken as the origin, but it is not essential. The coordinates \( (x, y, z) \) of any point \( P \) in space are shown in the figure. These are the position coordinates of an event at \( P \). Often, we consider the fourth coordinate \( t \) which represents the time of occurrence of the event. A reference frame with four coordinates \( (x, y, z; t) \) is called a space-time frame.

Is earth an inertial frame of reference?

Earth rotates around its axis as well as around the sun. In both these motions, centripetal acceleration is present. Therefore, strictly speaking, the earth or any frame of reference fixed on the earth cannot be taken as an inertial

frame. However, as we are dealing with speeds \( \approx 3 \times 10^8 \, \text{ms}^{-1} \) (speed of light) and the speed of the earth is only about \( 3 \times 10^4 \, \text{m/s} \), therefore when small time intervals are involved, effects of rotation and revolution of the earth can be ignored. Furthermore, this speed of earth can be assumed to be constant. Hence, the earth or any other frame of reference set up on earth can be taken as an approximately inertial frame of reference.

On the contrary, a frame of reference which is accelerated or decelerated is a non-inertial frame.

Note: Newton’s laws are not true in all reference frames. They hold only for inertial reference frames (i.e., unaccelerated frames).