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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

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

rotational kinematics equations

The average angular velocity of the body for a finite time interval is given by

rotational kinematics examples

 The unit of Angular Velocity is radian per second (rad/s).

The instantaneous angular velocity is defined as

                        Physics

 rotational kinematics problems and solutions
 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.

 Torque
     

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

rotational motion

        

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

rotational motion definition

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)

rotational motion problems

       

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

Constant Angular Acceleration Model  rigid body

 

Table 1   Analogy Between Rotational Kinematics and Linear Kinematics

equations of rotation kinematics


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

Linear Kinematics

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

angular displacement

rotational kinematics experiment

 

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

average angular velocity

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Rotational Kinematics

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