PHYSICS As LEVEL(FORM FIVE) NOTES – ROTATION OF RIGID BODIES(1)

• In other words, objects do not change their rotational motion unless acted upon by some not external torque.
• It is also called moment of inertia.

• It is denoted by I
• The greater the difficulty in starting or stopping , the greater is the moment of inertia of the body about that axis and vice-versa.
• A body rotates under the action of a net external torque.

………..mn situated at perpendicular distances …………… respectively from the axis of rotation yy’.

Initial angular velocity.

Final angular velocity after time t.

Uniform angular acceleration.

Angular displacement after time t.

1. To derive

At

2. To derive

From

At

From

Also

When

Consider a rigid body rotating about the axis with an angular speed as shown in figure.2 below

Suppose the body is made up of a large number of particles of masses m1, m2,m3 situated at perpendicular distances ……… respectively from the axis of rotation.

Although each particle within the body has the same angular speed , the velocity of each particle depends upon particles position with respect to axis of rotation.

When a rigid body rotates about an axis, possesses K.E called the kinetic energy of a rotating body, and its rotational kinetic energy denoted by ().

Let us now find the K.E of a rotating body. The particle of mass m1 follows a circular path of radius .

The rotational kinetic energy of the body is equal to the sum of rotational kinetic energies of all particles

.

is the moment of inertia of the body about the given axis of rotation

1. Note that both expressions for K.E are one half the products of property of body and square of velocity term.

2. If

From

2. Relation between torque and moment of inertia

Let the constant angular acceleration produced by the body be .

The angular acceleration () of all the particles of the body will be the same but the linear acceleration (a) of each particle will depend upon the particles position with respect to the axis of rotation.

The particle of mass m1 follows a circular path of radius r1. The magnitude of the linear acceleration of this particle a1 = r1 .

Similarly the magnitude of torque on the particles of masses m2,m3 ….. … .. are m2r22, m3 r32………… respectively.

The magnitude of the total torque on the body is just the sum of individual torques on the particles.

But is the moment of inertia of the body about the axis of rotation

The torque is analogous to force F, the moment of inertia I is analogous to mass m and the angular acceleration is analogous to the linear acceleration

If

3. Relation between angular momentum and moment of inertia

The particle of mass m1 follows a circular path of radius about the axis of rotation.

But is the moment of the body about the axis of rotation

But Torque

Consider a rigid body rotating about an axis due to a constant applied .

If the body rotates through a small change angle in small time then small work done on the body is

It is very important to keep in mind that it is – the product of moment of inertia and angular velocity that remains constant and not the angular velocity ω.

Imagine, a small infinitesimally element of mass “” and length “” at a distance “” from the axis of rotation passing through the middle of the uniform rod.

The moment of inertia of the imaginary small infinitesimally element a distance from the axis of rotation is given by

2. Moment of inertia of hollow Solid cylinder (ring)

Consider a small infinitesimal element (shell) at a distance r and having thickness,

3. Moment inertia of a solid cylinder (Disc)

Area at any radius from the centre of the circle

4. Moment of inertia of a sphere

Consider a spherical rigid body and a particle at a distance from the centre