Rotation of rigid body

Every particle of the rigid body in rotation undergoes circular motion irrespective of its shape. The centers of the circular paths described by them lie on the axis of rotation. It should be noted that the different particles, constituting rigid body, have different linear velocities, but same angular velocity. It means that each particle traverses same angle in a given time. The linear velocity of a particle is related to angular velocity as :

v = ω r v = ω r

where "r" is the perpendicular distance of the particle's position from the axis of rotation.

Figure 1: Each particle of the body follows a circular path about axis in pure rotational motion.

Pure rotational motion

It is clear from the relation that particle closer to the axis (smaller "r") will have lesser linear velocity than the one away from the axis (greater "r").

However, each of the particle of the body undergoes same angular displacement (θ), velocity (ω) and acceleration (α) in the case of rotation of rigid body about a fixed axis. In other words, we can say that the angular attributes of motion of the rigid body in rotation are uniquely (single valued) defined as in the case of a particle.

This situation is analogous to pure translational motion of rigid body, in which each particle constituting the body has same linear velocity and acceleration. For this reason, the motion of a particle or a rigid body in pure translational motion is governed by the same Newton's second law.

This suggests that the relationship between torque and resulting change in angular acceleration for rigid body in pure rotational motion should also be governed by same relation as that for a particle. Hence, governing relation of motion for rigid body is given as :

However, we need to evaluate moment of inertia of the rigid body appropriately so that it represents the inertia of the body to external torque (cause). In the next section, we shall drive the basic expression of moment of inertia for the rigid body.

Moment of inertia of a system of particles

Moment of inertia of a particle about fixed axis is given by :

The moment of inertia of the system of discrete particles is equal to the sum of moments of inertia of individual particles,

where " m i m i " is the mass of "i" th particle at linear distance " r i r i " from the axis of rotation.

Example 1

Problem : Three particles each of mass "m" are situated at the vertices of an equilateral triangle OAB of length "a" as shown in the figure. Calculate moment of inertia (i) about an axis passing through "O" and perpendicular to the plane of triangle (ii) about axis Ox and (iii) about axis Oy.

Figure 2: Three particles each of mass "m" are situated at the vertices of an equilateral triangle.

Moment of inertia

Solution : We shall make use of the formulae of moment of inertia for discrete particles in each of the cases :

(i) about an axis passing through "O" and perpendicular to the plane of triangle

Here, distances of three particles from the axis are :

r O = 0 ; r A = a ; r B = a r O = 0 ; r A = a ; r B = a

The moment of inertia about "O" and perpendicular to the plane of triangle is :

I = ∑ m i r i 2 = m o r o 2 + m A r A 2 + m B r B 2 I = ∑ m i r i 2 = m o r o 2 + m A r A 2 + m B r B 2

⇒ I = m x 0 + m a 2 + m a 2 = 2 m a 2 ⇒ I = m x 0 + m a 2 + m a 2 = 2 m a 2

(ii) about axis Ox

Here distances of three particles from the axes are :

r O = 0 ; r A = √ 3 a 2 ; r B = 0 r O = 0 ; r A = √ 3 a 2 ; r B = 0

I = ∑ m i r i 2 = m o r o 2 + m A r A 2 + m B r B 2 I = ∑ m i r i 2 = m o r o 2 + m A r A 2 + m B r B 2

⇒ I = m x 0 + m ( √ 3 a 2 ) 2 + m x 0 = 3 m a 2 4 ⇒ I = m x 0 + m ( √ 3 a 2 ) 2 + m x 0 = 3 m a 2 4

(iii) about axis Oy

Here distances of three particles from the axes are :

r O = 0 , r A = a 2 , r B = a r O = 0 , r A = a 2 , r B = a

I = ∑ m i r i 2 = m o r o 2 + m A r A 2 + m B r B 2 I = ∑ m i r i 2 = m o r o 2 + m A r A 2 + m B r B 2

⇒ I = m x 0 + m a 2 2 + m x a 2 = 5 m a 2 4 ⇒ I = m x 0 + m a 2 2 + m x a 2 = 5 m a 2 4

This example illustrates how choice of axis changes moment of inertia i.e. the inertia of the system of particles to rotation. This is expected as change in the reference axis actually changes distribution of mass about axis of rotation.

Moment of inertia of rigid body

Rigid body is a continuous aggregation of particles. We, therefore, need to modify the summation in the expression of moment of inertia by integration as :

Evaluation of this integral for a given body is a separate task in itself. It is mathematically possible to evaluate this integral for bodies of regular shape. It would, however, be very difficult to evaluate the same for irregularly shaped rigid body. In such cases, it is pragmatic to resort to experimental methods to calculate moment of inertia. Mathematical evaluation of moment of inertia even for regularly shaped bodies would require specialized analysis and evaluation.

Two theorems, pertaining to moment of inertia, are of a great help in the mathematical evaluation of moment of inertia of regularly shaped bodies. They are (i) parallel axes theorem and (ii) perpendicular axes theorem. These theorems extend the result of moment of inertia of basic geometric forms of rigid bodies to other axes and body forms. We shall cover these aspects and shall evaluate moment of inertia of certain important geometric rigid bodies in a separate module.

Kinetic energy of rigid body in rotation

Kinetic energy of a particle or body represents the form of energy that arises from motion. We are aware that kinetic energy of a particle in translation is given by the expression :

In pure rotation, however, the rigid body has no "over all" translation of the body. However, the body in rotation must have kinetic energy as it involves certain motion. A closer look on the rotation of rigid body reveals that though we may not be able to assign translation to the rigid body as a whole, but we can recognize translation of individual particles as each of them rotate about the axis in circular motion with different linear speeds. The speed of a particle is given by :

Figure 3: Each particle of the body follows a circular path about axis in pure rotational motion.

Pure rotational motion

v i = ω r i v i = ω r i

Thus, kinetic energy of an individual particle is :

K i = 1 2 m i v i 2 K i = 1 2 m i v i 2

where " K i K i " is the kinetic energy of "i" th particle having a speed " v i v i ". In terms of angular speed, the kinetic energy of an individual particle is :

K i = 1 2 m i ( ω r i ) 2 K i = 1 2 m i ( ω r i ) 2

Now, the kinetic energy of the rigid body is sum of the kinetic energies of the particles constituting the rigid body :

K = ∑ K i = ∑ 1 2 m i ω 2 r i 2 K = ∑ K i = ∑ 1 2 m i ω 2 r i 2

We note here that angular speeds of all particles constituting the body are same. Hence, the constant "1/2" and " ω 2 ω 2 " can be taken out of the summation sign :

K = 1 2 ω 2 ∑ m i r i 2 K = 1 2 ω 2 ∑ m i r i 2

However, we know that :

I = ∑ m i r i 2 I = ∑ m i r i 2

Combining two equations, we have :

This is the desired expression of kinetic energy of a rigid body rotating about a fixed axis i.e. in pure rotational motion. The form of expression of the kinetic energy here emphasizes the correspondence between linear and angular quantities. Comparing with the expression of kinetic energy for translational motion, we find that "moment of inertia (I)" corresponds to "mass (m)" and "linear speed (v)" corresponds to "angular speed (ω)".

We can also interpret the result obtained above from a different perspective. We could have directly inferred that expression of kinetic energy in rotation should have an equivalent form as :

K (Kinetic energy) = 1 2 x (inertia) x (speed) 2 K (Kinetic energy) = 1 2 x (inertia) x (speed) 2

In rotation, inertia to the rotation is "moment of inertia (I)" and speed of the rigid body is "angular speed (ω)". Substituting for the quantities, we have the expression for kinetic energy of rigid body in rotation as :

K = 1 2 I ω 2 K = 1 2 I ω 2

Comparing this equation with the expression of the sum of kinetic energy of individual particles as derived earlier :

K = 1 2 ω 2 ∑ m i r i 2 K = 1 2 ω 2 ∑ m i r i 2

Clearly,

I = ∑ m i r i 2 I = ∑ m i r i 2

This conclusion, thus, clearly establishes that the expression as given by ∑ m i r i 2 ∑ m i r i 2 indeed represents the inertia of the rigid body in rotation.

Summary

1. Moment of inertia is the inertia of an object against any change in its state of rotation. This quantity corresponds to “mass”, which determines inertia of an object in translational motion.

2. Moment of inertia is defined for rotation about a fixed axis for a particle, a system of particles and a rigid body.

3. Expressions of Moment of inertia

(i) For a particle

I = m r 2 I = m r 2

(ii) For a system of particles

I = ∑ m i r i 2 I = ∑ m i r i 2

(iii) For a rigid body

I = ∫ r 2 ⅆ m I = ∫ r 2 ⅆ m

4. Moment of inertia, also referred in short as MI, is a scalar quantity. The directional positions (angular positions) of the particles/objects with respect to axis of rotation does not matter. The unit of MI is kg - m 2 kg - m 2 .

5. Objects in rotation have rotational kinetic energy of rotation due to rotational motion of individual particles, constituting the object.

6. Object in pure rotation has only rotational kinetic energy i.e. no translational kinetic energy is involved.

7. The expression of rotational kinetic energy is given by :

K = 1 2 I ω 2 K = 1 2 I ω 2