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ROTATION OF RIGID BODIES

Linear inertia

Is the tendency of a body to resist change in its linear velocity.

In other words; objects do not change their state of linear motion unless acted upon by some not external force.

Rotational inertia

Is the tendency of a body to resist change in its angular velocity.

- 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.

CONCEPT OF MOMENT OF INERTIA

Moment of inertia of a body about an axis is a measure of the difficulty in starting, stopping or changing rotation of the body about that axis.

- 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.

The Greater the moment of inertia of a body about an axis of rotation, the greater is the torque required to rotate or stop or change rotation of the body that axis and vice-versa.

MOMENT OF INERTIA OF A RIGID BODY

Consider a rigid body rotating about the axis yy-1 with an angular speed ω as shown in figure 1 below.

Figure. 1

Suppose the body is made up of a large number (n) of small particles of masses m1,m2,m3

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

As the body rotates, each particle of the body follows a circular path around the axis.

Although each particle of the body has the same angular speed ω, the linear velocity (v) of each particle depends upon particles distance from the axis of rotation.

Thus particle of mass m1 follows a circular path of radius r1. The linear velocity of this particle is v1

Rotational kinetic energy of the particles of mass m1

The total kinetic energy Kr of the rotating body is the sum of the kinetic energies of all the particles of which the body is composed.

Hence the total K.E of rotating body

Moment of inertia of a rigid body about a given axis of rotation

Is the sum of the products of the masses of its particles and the squares of their respective perpendicular distance from the axis of rotation.

The moment of inertia of a body about an axis of rotation is directly proportional to the total mass of the body.

The more massive the body, the more difficult will be to start its rotational motion or stop it from rotating.

For a given mass, the moment of inertia of a body depends upon the distribution of the mass from the axis of rotation. The larger the distance of the mass from the axis rotation the larger will be its moment of inertia.

The moment of inertia plays the same role in rotational motion as mass plays in translational motion.

RADIUS OF GYRATION

Radius of gyration Is the distance from the given axis of rotation at which if whole mass of the object were supposed to be concentrated the moment of inertia would be the same as with the actual distribution of mass.

The radius of gyration is denoted by the symbol K.

T The moment of inertia of a body of mass M and radius of gyration K is given by,

For example, the moment of inertia of a thin rod of mass M and length L about an axis through its centre and perpendicular to its length is

The radius of gyration is a measure of the distribution of mass of a body relative to a given axis of rotation.

A large radius of gyration means that, on the average, the mass is relatively far from the given axis of rotation.

The SI unit of radius of gyration is the metre (m)

Suppose a body consist of n particles each of mass m, then total mass of the body is M.

Therefore radius of gyration is the root mean square distance of the various particles of the body from the axis of rotation.

EQUATIONS OR UNIFORMLY ACCELERATED ROTATIONAL MOTION

Consider a rigid body rotating about a given axis with uniform angular acceleration.

Let

Initial angular velocity.

Final angular velocity after time t.

Uniform angular acceleration.

Angular displacement after time t.

- To derive

From

At

For linear motion

v = u + at

- To derive

From

At

For the linear motion

From

Also

When

For linear motion

v2 = u2 + 2as

MOTION OF A RIGID BODY AND MOMENT OF INERTIA

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

Figure. 2

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.

As the body rotates, each particle within the body follows a circular path around the axis.

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.

1. Relation between rotational kinetic energy and moment of inertia.

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 magnitude of the linear or tangential velocity of the particle on this circle is v1

Similarly, the rotation kinetic energy of particles of masses

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

Thus the rotational K.E of a body is equal to half of the product of the moment of inertia of the body and the square of the angular velocity of the bo

dy about a given axis of rotation.

dy about a given axis of rotation.

We also know that kinetic energy K of a body for linear motion is,

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

We know that linear velocity v is an analogue of angular velocity ω in rotational motion

Therefore, mass (m) of the body is an analogue of moment of inertia I of the body in rotation motion.

Hence we arrive at a very important conclusion that moment of inertia plays the same role in rotational as the mass (m) plays in linear motion.

- If

From

Thus, the moment of inertia of a rigid body about a given axis of rotation is numerically equal to twice the rotational K.E of the body when rotating with unit angular velocity about that axis.

##### 2. Relation between torque and moment of inertia

Suppose a body rotates about an axis under the action of a constant to torque τ .

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 .

If F1 is the net external force acting on this particle

The magnitude of torque due to this force on this particle

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

By right hand rule, the torques on all the particles act in the same direction.

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

This is basic relation for rotational motion

It is analogous to Newton’s second law of linear motion

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

Hence the moment of inertia of a body about a given axis is equal to the torque required to produce unit angular acceleration in the body about that axis.

##### 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.

Therefore, the magnitude v1 of linear velocity of this particle is

The magnitude of the angular momentum of this particle about the axis of rotation

= linear momentum x radius r1

= m1 v1 x r1

Similarly, the magnitude of angular momentum of particles of masses m2,m3 ….. … ..

are m2r22ω, m3r32 ω …………….. respectively.

By right hand rule, the angular momentum of all the particles L of the rigid body point in the same direction, parallel to the angular velocity vector.

Therefore, the magnitude of total angular momentum L of the body about the axis of rotation is just the sum of individual momentum of the particles.

L = m1r12ω + m2r22ω + m31r32ω + …..

= (m1r12 + m2r22 + m3r32)ω

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

The SI units of L is kgm2s-1

4. Relation between torque and angular momentum.

The magnitude of angular momentum of a body about an axis is

Differentiating both sides with respect to t we get

But Torque

Then

Thus the torque acting on a body is equal to time rate of change of angular momentum of the body.

Power in Rotational Motion

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

Thus power of a rotating body is equal to the product of torque and angular velocity.

Work-energy theorem in rotational motion

In linear motion, the work-energy theorem states that the net work done on a body by the external force is equal to the change in body’s linear kinetic energy.

Similar, in rotational motion, the work done by a torque in rotating the body about an axis is equal to the change in body‘s rotation kinetic energy.

Therefore, work-energy theorem in rotational motion may be stated as under “The net work-done by the external torque in rotating a rigid body about an axis is equal to the change in body’s rotational K.E”.

LAW OF CONSERVATION OF ANGULAR MOMENTUM

If not net external torque acts on a body or system, its angular momentum remains constant in magnitude and direction.

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.

MOMENT OF INERTIA AND RADIUS OF GYRATION OF DIFFERENT BODIES WITH REGULAR GEOMETRIC SHAPES

1. Moment of inertia of a uniform rod

(a) About an axis through its centre

Consider a uniform rod of total mass M and length L

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.

Figure. 3

By similarity

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

The total moment of inertia for the whole uniform rod is obtained by integrating the above expression

Total moment of inertia for the whole rod is

Its Radius of gyration

But is the moment of inertia of a uniform rod about an axis through its centre

Then

(b) Moment of inertia about an axis at one end

Figure 4

By similarity

Recall

dl = dmx2

By integrating both sides

- Moment of inertia of hollow Solid cylinder (ring)

Consider a hollow solid cylinder of uniform density is related about an axis passing through the center along the length L of the cylinder.

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

then dV = area x thickness

The mass of shell is given by

The moment of inertia for a shell

But the mass of the cylinder is given

Then

Hence for hollow cylinder (ring) the moment of inertia is given by

But if the thickness of the cylinder is very small

I = MR2 is the moment of inertia for thin hoop.

Its radius of gyration

From

MR2 = MK2

K2 = R2

Therefore, K = R

- Moment inertia of a solid cylinder (Disc)

Figure. 6

Area at any radius from the centre of the circle

Differentiate

By similarity

But dI = x2dm

Its radius of gyration

From

- Moment of inertia of a sphere

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

Figure. 7

Volume of a small element

Mass of a sphere

Moment of inertia of a shell

PARALLEL AXES THEOREM

The parallel axes theorem state that moment of inertia of a rigid body about any given axis is equal to its moment of inertia about a parallel axis through its centre of mass plus the product of the mass of the body and the square of the perpendicular distance between the two parallel axes.

2

I= IG + Mh

This is a very simple and useful theorem for objects of regular shapes e.g. rod, sphere, disc. Application of parallel axes theorem