Maharashtra Board Class 12 Physics Chapter 1 Rotational Dynamics PDF Download

1. Rotational Dynamics

Can You Recall?

1. What is circular motion?

2. What is the concept of centre of mass?

3. What are kinematical equations of motion?

4. Do you know real and pseudo forces, their origin and applications?

1.1 Introduction

Circular motion is an essential part of our daily life. Every day we come across several revolving or rotating objects. During revolution, the object undergoes circular motion about some point outside the object. During rotation, the motion is about an axis of rotation passing through the object.

1.2 Characteristics of Circular Motion

1) It is an accelerated motion. The direction of velocity changes at every instant. So it is an accelerated motion.

2) It is a periodic motion. During the motion, the particle repeats its path along the same trajectory. Thus, the motion is periodic.

1.2.1 Kinematics of Circular Motion

To describe circular motion, we use angular displacement \(\theta\), angular velocity \(\omega = \frac{d\theta}{dt}\), and angular acceleration \(\alpha = \frac{d\omega}{dt}\). These are analogous to displacement, velocity, and acceleration in translational motion.

The tangential velocity is given by \(\vec{v} = \vec{\omega} \times \vec{r}\) where \(\vec{\omega}\) is the angular velocity. The magnitude of \(\vec{v}\) is \(v = \omega r\).

The direction of \(\vec{\omega}\) is along the axis of rotation. To find the direction, curl the fingers of the right hand along the sense of rotation. The outstretched thumb gives the direction of \(\vec{\omega}\).

If T is the period of circular motion and n is the frequency, then \(\omega = 2\pi n = \frac{2\pi}{T}\).

Uniform Circular Motion

During circular motion, if the speed of the particle remains constant, it is called Uniform Circular Motion (UCM). Only the direction of velocity changes. The velocity is always tangential to the path. The centripetal or radial acceleration is \(\vec{a}_r = -\omega^2\vec{r}\). For UCM, its magnitude is constant: \(a = \omega^2 r = \frac{v^2}{r} = v\omega\). It is always directed towards the centre of the circular motion.

Teacher's Note

Circular motion happens in many Indian games and activities. Think of a stone tied to a string being whirled around - this is circular motion.

Exam Trick

Remember: Centripetal acceleration always points towards the centre. Like gravity always pulls towards Earth, centripetal force always pulls towards the centre of the circle.

Points to Remember

Circular motion is when an object moves in a circle around a fixed point.
The direction of velocity is always along the tangent to the circle.
Centripetal acceleration always points towards the centre.
For uniform circular motion, speed is constant but direction keeps changing.
Angular velocity and linear velocity are related by \(v = \omega r\).

Illustration

Circular motion of any particle of a fan rotating uniformly is a good example.

Non-Uniform Circular Motion

When a fan is switched ON or OFF, the speeds of particles go on increasing or decreasing. However, their directions are always tangential to their circular paths. During this time, it is a non-uniform circular motion.

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