# Inertial and Non-inertial Frames

A reference frame is used to describe motion (position, velocity, acceleration) of an object. It is a collection of three non-colinear points whose mutual distances is constants.

A reference frame can have many coordinate systems. A coordinate system is defined by (i) a point (origin) fixed in the reference frame and (ii) three orthogonal directions.

Let us learn these concepts through an example. Let us consider a reference frame S defined by three non-colinear points (say A, B and C) on your table. In reference frame S, let us define a coordinate system with

1. origin at the point A
2. one direction along AB, another direction is perpendicular to AB but lies in the plane of the table and, third direction is perpendicular to the plane of the table.

An observer (you) sitting at the point A in this reference frame (and coordinate system) can describe the motion of an ant crawling on the table or an insect flying in the room.

There are two types of reference frames: inertial and non-inertial.

You need to be careful while defining inertial frame. Suppose you are observing a free particle (with no force acting on it). If you find particle's acceleration to be zero then you are in an inertial frame. A frame moving with a constant velocity relative to an inertial frame is also inertial.

Recall the statement of Newton's first law "if a body is at rest or moving at a constant speed in a straight line, it will remain at rest or keep moving in a straight line at constant speed unless it is acted upon by a force." In which frame "rest" or "constant speed" is defined? In the inertial frame. In essence, Newton's first law defines an inertial frame.

A frame which is not inertial is a non-inertial frame. A frame accelerating relative to an inertial frame is non-inertial. A frame rotating relative to an inertial frame is non-inertial. Rotating frames are non-inertial frames.

The formula $\vec{F}=m\vec{a}$ (Newton's second law) considers acceleration $\vec{a}$ in inertial frame. You need to add additional terms (e.g., centrifugal, coriolis etc.) if acceleration is in non-inertial frame (e.g., rotating frame). These additional terms are called pseudo forces.

## Problems from IIT JEE

Problem (IIT JEE 1986): A reference frame attached to the earth,

1. is an inertial frame by definition.
2. cannot be inertial frame because the earth is revolving around the sun.
3. is an inertial frame because Newton's laws are applicable in this frame.
4. cannot be an inertial frame because the earth is rotating about its own axis.

Solution: The inertial frame is defined by Newton's first law of motion. Suppose there is a particle with no physical force (gravitational, electromagnetic, weak, or nuclear) acting on it. By Newton's first law, this particle is at 'rest' or in 'uniform motion'. But 'rest' or 'uniform motion' are defined w.r.t. a reference frame. The reference frame in which this particle is at 'rest' or in 'uniform motion' is called an inertial frame.

A frame having uniform motion w.r.t. another inertial frame is also inertial. An accelerating or rotating frame is non-inertial. Thus, the frame attached to the earth is non-inertial because this frame is rotating due to spinning of the earth about its axis as well as due to revolution of the earth around the sun. The readers are encouraged to find the angular velocity of this frame due to spinning and revolution of the earth. Hint: $\omega_\text{spin}=15\;\mathrm{deg/hr}$, $\omega_\text{revolution}=0.04\;\mathrm{deg/hr}$.