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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.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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 EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)……………EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1) 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
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Rotational kinetic energy of the particles of mass m1
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Hence the total K.E of rotating body
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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,
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)Initial angular velocity.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)Final angular velocity after time t.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)Uniform angular acceleration.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)Angular displacement after time t.
  1. To derive EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
From
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
At EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
For linear motion
v = u + at
  1. To derive EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
From EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
At EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
For the linear motion
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
From EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Also EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
When EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
For linear motion
v2 = u2 + 2as
MOTION OF A RIGID BODY AND MOMENT OF INERTIA
Consider a rigid body rotating about the axis EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)with an angular speed EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1) as shown in figure.2 below
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Figure. 2
Suppose the body is made up of a large number of particles of masses m1, m2,m3 situated at perpendicular distances EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)……… 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 EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1), 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 (EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)).
Let us now find the K.E of a rotating body. The particle of mass m1 follows a circular path of radius EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1).
The magnitude of the linear or tangential velocity of the particle on this circle is v1
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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 EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
.EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
is the moment of inertia of the body about the given axis of rotation EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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,
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
  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 EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
From EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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 EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1).
The angular acceleration (EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)) 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 EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1) .
If F1 is the net external force acting on this particle
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
The magnitude of torque due to this force on this particle
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Similarly the magnitude of torque on the particles of masses m2,m3 ….. … .. are m2r22EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1), m3 r32EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)………… respectively.
By right hand rule, the torques on all the particles act in the same direction.
The magnitude of the total torque EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)on the body is just the sum of individual torques on the particles.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
But EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1) is the moment of inertia of the body about the axis of rotation
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
 This is basic relation for rotational motion
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
It is analogous to Newton’s second law of linear motion
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
The torque is analogous to force F, the moment of inertia I is analogous to mass m and the angular acceleration EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1) is analogous to the linear acceleration
If EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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 EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1) about the axis of rotation.
Therefore, the magnitude v1 of linear velocity of this particle is
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
But EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1) is the moment of the body about the axis of rotation
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Differentiating both sides with respect to t we get
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
But Torque EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Then
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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 EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1).
If the body rotates through a small change angle EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1) in small time EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)then small work EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)done on the body is
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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”.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
It is very important to keep in mind that it is EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1) – the product of moment of inertia and angular velocity that remains constant and not the angular velocity ω.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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 “EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)” and length “EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)” at a distance “EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)” from the axis of rotation passing through the middle of the uniform rod.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Figure. 3
By similarity
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
The moment of inertia of the imaginary small infinitesimally element a distance EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)from the axis of rotation is given by
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
The total moment of inertia for the whole uniform rod is obtained by integrating the above expression
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Total moment of inertia for the whole rod is
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Its Radius of gyration
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
But is the moment of inertia of a uniform rod about an axis through its centre
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Then
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
(b) Moment of inertia about an axis at one end
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Figure 4
By similarity
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Recall
dl = dmx2
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
By integrating both sides
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
  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.
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Consider a small infinitesimal element (shell) at a distance r and having thickness, EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
then dV = area x thickness
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
The mass of shell is given by
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
The moment of inertia for a shell
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
But the mass of the cylinder is given
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Then
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Hence for hollow cylinder (ring) the moment of inertia is given by
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
But if the thickness of the cylinder is very small
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
I = MR2 is the moment of inertia for thin hoop.
Its radius of gyration
From
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
MR2 = MK2
K2 = R2
Therefore, K = R
  1. Moment inertia of a solid cylinder (Disc)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Figure. 6
Area at any radius EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)from the centre of the circle
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Differentiate
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
By similarity
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
But dI = x2dm
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Its radius of gyration
From
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
  1. Moment of inertia of a sphere
Consider a spherical rigid body and a particle at a distance EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1) from the centre
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Figure. 7
Volume of a small element
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Mass of a sphere
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
Moment of inertia of a shell
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)
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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EcoleBooks | PHYSICS As LEVEL(FORM FIVE) NOTES - ROTATION OF RIGID BODIES(1)

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