Rigid Body : Ideally, a body is said to be a rigid body when it has a perfectly definite shape and size and the distances between particles of such a body do not change while applying an external force on it

e.g. A wheel can be considered as rigid body by ignoring change in its shape.

The rigid body may have two kinds of motion as follows:

1. Pure Translational Motion :In pure translational motion, at any instant of time all particles of the body have the same velocity. e.g. A block of square shape moving on a plane surface.
2. Pure Rotational Motion : In pure rotational motion, every particle of the rigid body moves in circles of different radii about a fixed line, which is known as axis of rotation. e.g. An oscillating fan.

Some rigid bodies may have a rolling motion, which is the combination of rotation and translation. e.g. A cylinder rolls down on inclined plane.

Centre of Mass : A point  at which the entire mass of the body or system of bodies is supposed to be concentrated is known as the centre of mass.

Let us consider the system of two particles of masses m₁ and m₂, their distance x₁ and x₂ respectively from origin O.

Then, the centre of mass of the system at a point, which is at distance of x_CM, from O is given by

Center of Mass x_CM= \( \frac{m_1x_1+m_2x_2}{m_1+m_2}\)

Where M=m₁+m₂ = Total mass of system

Moment of Inertia of large Rigid Body :

Let us consider a body of n particles of masses m₁, m₂,..., m_n, with r₁, r₂,..., r_n as their respective distances from axis of rotation as shown in figure.

Then, moment of inertia of the body is given by

I= m₁r₁² + m₂r₂² +... + m_nr_n²

OR

 \(I=\sum \limits_{i=1}^{n}m_ir_i^2\)

Physical Significance of Moment of Inertia

The unwillingness of a body to change its state of rest or of uniform motion along a straight line is called as Inertia.

The greater is the mass of a body, the greater is the force required to change its state of motion alongs a straight line.

Thus the mass of a body is a measure of its inertia.

Similarly, the moment of intertia of a body about an axis of rotation resists a change in its rotational motion. The greater of the moment of intertia of a body the greater is the torque required to change its state of rotation.

Thus, moment of inertia plays the same role in rotational motion as mass plays in translational motion,

Theorem of Parallel Axes

It states that, “The moment of inertia of a body about an axis is equal to sum of moments of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of distance between the two parallel axes”.

Let I be the moment of inertia of the body about an axis passing through point O. Take another parallel axis of rotation passing through the centre of mass of the body. Let I_CM be the moment of inertia of the body about the axis passing through the centre of mass the distance between two parallel axis is h. Then, according to theorem of parallel axes

Moment of inertia, I = I_CM + Mh²

where, M = mass of lamina

h = distance between two parallel axes.

Theorem of Perpendicular Axes :

It states that,

“The moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two mutually perpendicular in its own pane and intersecting each at the point where the perpendicular axis passes through the body (lamina)”.

Let I_XX, I_YY and I_ZZ, be the moment of inertia of the lamina about OX,OY and OZ-axes respectively. Then, theorem of perpendicular axis gives

Moment of inertia  I_ZZ = I_XX + I_YY

Perpendicular axes showing a planar body, where x and y are perpendicular axes and Z is perpendicular to the plane.

Moment of Inertia in Some Standard Cases:

Torque of a Force about the Axis of Rotation

Consider a rigid body rotating about a fixed axis AB. Let F be a force acting on the body at point P. Take the origin O somewhere on the axis of rotation. The torque of F about O is defined as,

Force F acting at a point P on the rigid body

τ = r x F  or τ = Iα

where

I = moment of inertia of the body about the axis of rotation

α = angular acceleration of the body

Torque produces only rotational motion of the body.

Work done by a torque τ in turning the body through a small angle dθ is dW = τdθ.

Or W = τθ.

Power of the torque, P = τ (dθ/dt) = τ.ω

Law of Conservation of Angular Momentum

We know that, τ_ext = dL/dt

If τ_ext =0, then dL/dt =0 or L must be a constant. Therefore, the law of conservation of angular momentum states that

“the angular momentum of a rotating body is conserved or remains constant, if the resultant external torque acting on the body is zero”.

As L = Iw the law of conservation of momentum leads us to the conclusion.

For an isolated system, Iw = constant

or I₁w₁ = I₂w₂                                

This principle is often made use by gymnast swimmers, Circus acrobats and ballet dancers, etc.

If external torque is acting on a body.

dL/dt = τ_ext

τ_ext dt = dL

Integrating this

J=L₂-L₁

Thus, the change in angular momentum is equal to the angular impulse of the torque.

Pure Rolling Motion : If the relative velocity of the point of contact (between body and platform) is zero, then the rolling motion is said to be pure rolling motion.

For pure rolling motion, v_AB =0

i.e. (v-Rω)-v₀ =0

v-Rω= v₀

• If the platform is stationary, i.e. v₀ = 0, then for pure rolling motion, v = Rw.
• In general, friction is responsible for pure rolling motion. In pure rolling motion, friction is static in nature.
• In pure rolling motion, friction is non-dissipative in nature, i.e. work done by frictional force is zero, this is because of the fact that point of contact is relatively at rest.
• In case of pure rolling motion, velocity of different particles of the body is as shown below.

For above body, total kinetic energy is given by

K = K_R + K_T = \(\frac{Iω^2}{2} + \frac{mv^2}{2} \)

= \(\frac{Iω^2}{2} + \frac{m(Rω)^2}{2} \)

= \(\frac{(1+mR^2)ω^2}{2}  \)

= \(\frac{I_Pω^2}{2} \)

Total kinetic energy, K = \(\frac{1}{2}I_Pω^2 \)

where, I_P =I+mR² (the moment of inertia of a body passing through axis is at point of contact).

Rolling Motion on an Inclined Plane : We consider an inclined plane of q inclination on which a body performs pure rolling motion. At any instant t, the body is at a height h from the horizontal surface.

We suppose that body is released from height h,

Rolling motion on an inclined plane

Comparison of Various Motions of a Body on an Inclined Plane :

1) Rolling Motion :

Velocity \(v_r = \sqrt{\frac{2gh}{β}} = \sqrt{\frac{2gh}{1+\frac{K^2}{R^2}}}\)

Acceleration \(a_r = \frac{gsinθ}{β} =\frac{gsinθ}{1+\frac{K^2}{R^2}}\)

Time of Descend \(T_r = \frac{1}{sinθ}\sqrt{\frac{2hβ}{g}}\)

2) Sliding Motion :

Velocity \(v_s = \sqrt{2gh}\)

Acceleration \(a_s = gsinθ\)

Time of Descend \(T_s = \frac{1}{sinθ}\sqrt{\frac{2h}{g}}\)

3) Falling Motion :

Velocity \(v_f = \sqrt{2gh}\)

Acceleration \(a_f = g\)

Time of Descend \(T_f = \sqrt{\frac{2h}{g}}\)

Impure rolling motion : In impure rolling motion, the point of contact of the body with the platform is not relatively at rest with respect to platform on which it is performing rolling motion, as a result sliding occurs at the point of contact, or impure rolling motion, V_AB ≠ 0, i.e. v— Rω ≠ v₀

If platform is stationary, i.e. v₀ = o, then condition for pure rolling motion is v ≠ Rω.

Here v ≠ Rω.. so a ≠ Rα