# Rotational Kinematics

In this chapter we will study the rotational motion of a rigid body about a fixed axis. A rigid body is defined as an object that has fixed size and shape. In other words, the relative positions of its constituent particles remain constant. In actual, a rigid body does not exist – it is an useful idealization. By the term fixed axis, we mean that the axis must be fixed relative to the body and fixed in direction relative to an inertial position.

## ROTATIONAL KINEMATICS

Consider a body of arbitrary shape rotating about a fixed axis ‘O’ as shown in figure(2). In a given interval all the particles lying on the line OA move to their corresponding positions lying on OB.

Although the particles of the body have different linear displacements, they all have the same Angular displacement q, which is given by

 The average angular velocity of the body for a finite time interval is given by  The unit of Angular Velocity is radian per second (rad/s). The instantaneous angular velocity is defined as It is the rate of change of the angular position q with respect to time. It is a vector quantity.  The direction of angular velocity is given by the right-hand rule. We hold the right hand such that when the fingers of the right hand curve in the sense of rotation, the thumb points in the direction of ω.

### Period and Frequency of Revolution

The period T is the time for one revolution and the frequency f is the number of revolutions per second (rev/s). The period and frequency are related as

Although all particles have the same angular velocity, their speeds increase linearly with distance from the axis of rotation.

The average angular acceleration is defined as

Angular acceleration is a vector quantity measured in rad/s2.

### The Constant Angular Acceleration Model

When the angular acceleration is constant, we can find the change in angular velocity by integrating equation (8)

A particle moving in a circular path with speed v has a centripetal (or radial) acceleration

### Illustration-1

A disc starts rotating with constant angular acceleration of p rad/s2 about a fixed axis perpendicular to its plane and through its centre.

(a)  Find the angular velocity of the disc after 4 s.

(b)  Find the angular displacement of the disc after 4 s and

(c)  Find number of turns accomplished by the disc in 4 s.

Solution

### Illustration-2

A wheel rotates with an angular acceleration given by a = 4at3 – 3bt2 , where t is the time and a and b are constants. If the wheel has initial angular speed w0, write the equations for the:

(i) angular speed                                 (ii) angular displacement.

Solution

### Illustration –3

A flywheel of radius 20 cm starts from rest, and has a constant angular acceleration  of 60 rad/s2.  Find

(a)  the magnitude of the net linear acceleration  of a point on the rim after 0.15 s

(b) the number of revolutions completed in 0.25 s

Solution

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