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 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.
1. To derive
From
At
For linear motion
v = u + at
1. 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.
We also know that kinetic energy K of a body for linear motion is,
1. 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.
1. 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
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
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
1. 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.
From
MR2 = MK2
K2 = R2
Therefore, K = R
1. 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
From
1. 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

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