Inertial and Non-inertial Frames of Reference Physics Notes

Frame of Reference:
The motion of a body is always described with reference to some well-defined coordinate system. This coordinate system is referred to as a frame of reference’.

In three-dimensional space, a frame of reference consists of three mutually perpendicular lines called ‘axis’ or ‘frame of reference’ meeting at a single point of origin. The coordinates of the origin are O (0, 0, 0) and that of any other point ‘P in space are P(x,y,z). The line joining the points O and P is called the position vector of the point P with respect to ‘O’.
Inertial and Non-inertial Frames of Reference Physics Notes 1

Inertial Frame of Reference:
A frame of reference that remains at rest or moves with a constant velocity with respect to the other frame of reference is called ‘Inertial Frame of Reference’. An inertial frame of reference is actually an unaccelerated frame of reference. Newton’s law of motion are valid in all inertial frames of reference. In this frame of reference, a body is not acted upon by external forces. All inertial frames of reference are equivalent for the measurement of physical phenomena.

Examples:

Our Earth.
A space shuttle moving with a constant velocity relative to the earth.
A rocket moving with a constant velocity relative, to the Earth.

Non-inertial Frame of Reference:
A frame of reference is said to be a non-inertial frame of reference when a body, not acted upon by an external force, is accelerated. In the non-inertial frame of reference, Newton’s law of motion are not valid unless we introduce a force, called pseudo force.

For example, A freely falling elevator may be taken as a non-inertial frame.
Is Earth an Inertial-Frame of Reference?
The Earth rotates around its axis and also revolves around the Sun. In both these motions, centripetal acceleration is present. Therefore, strictly speaking, the Earth or any frame of reference fixed on Earth can not be taken as an inertial frame. However, as we are dealing with speed ≈ ×10⁸m/s, (speed of light) and speed of Earth is only about 3 × 10⁴ m/s, therefore when small time intervals are involved, the effect of rotation and revolution of Earth can be ignored. Furthermore, this speed of the Earth can be assumed to be constant.

Hence, the Earth or any other frame of reference set up on the Earth can be taken as an approximately inertial frame of reference.
On the contrary, a frame of reference which is accelerated is the non-inertial frame.

Other examples of inertial frames of reference are:

A frame of reference remaining fixed w.r.t. Stars.
A spaceship moving in outer space, without spinning and with its engine cut off.

Solution of Problems in Mechanics by Concurrent Forces and Force Diagram
Equilibrium of Concurrent Forces:
A number of forces acting on the same point are called concurrent forces and these forces are said to be in equilibrium if their resultant is zero.

Let us consider two forces \(\vec{F}_{1}\) and \(\vec{F}_{2}\) acting at a point simultaneously. They will be in equilibrium if they are equal and opposite to each other, i.e.,
\(\vec{F}_{1}=-\vec{F}_{2}\)
or \(\vec{F}_{1}+\vec{F}_{2}\) = 0

Now, consider three concurrent forces \(\vec{F}_{1}\), \(\vec{F}_{2}\)and \(\vec{F}_{3}\) acting on a point O as shown in the figure.

The resultant of two forces \(\vec{F}_{1}\) and \(\vec{F}_{2}\) can be determined by using parallelogram law of vector addition which comes to be (\(\vec{F}_{1}+\vec{F}_{2}\)) as shown in the figure.

For equilibrium, \(\vec{F}_{3}\) must be equal and opposite to (\(\vec{F}_{1}+\vec{F}_{2}\))
i.e., (\(\vec{F}_{1}+\vec{F}_{2}\)) = –\(\vec{F}_{3}\)
or F₁ + F₂ + F₃ =0 …(1)
Inertial and Non-inertial Frames of Reference Physics Notes 2
In general if there are n forces, then at equilibrium,
\(\vec{F}_{1}+\vec{F}_{2}+\vec{F}_{3}+\ldots+\vec{F}_{n}\) …………..(2)

Free Body Diagram
To solve a typical problem in mechanics, we can use the following steps:

Draw a diagram depicting systematically the various parts of the assembly of bodies, supports, links, etc.
Select a convenient part of the assembly as one system.
Draw another diagram that shows the system and all the forces acting on the system. This diagram is called a ‘Free Body Diagram’ (FBD).
In a FBD, mention information about forces (magnitudes and directions).
If required, follow the same procedure for another choice of the system. In doing so, use third law of motion. For example, if in the FBD of A, the force on A due to Bis F, then in the FBD of B, the force on B due to A will be -F.