Kinematics of circular motion - Equations, Practice problems, FAQs

Sarvesh wakes up to look at the rotating fan on the ceiling above him. He is able to observe that all particles of the fan are rotating with the same angular velocity, and cover the same angle within the same time. Now he speeds up the fan using the regulator, he notices that its angular velocity of rotation increases gradually. He then moves over to his table on which he had kept a spinning toy. He gives it a spin and notices that the top spins at a certain speed when set into motion; but as time passes, it loses its speed; in other words it decelerates. The angular velocity of the toy gives an idea about the angle swept by the toy in a certain time interval. Let us say that the spin toy started out with an angular velocity of . In order to find what speed it has lost after some time, say seconds, knowledge of kinematics is needed: a branch of mechanics that deals with bodies in motion without considering the forces responsible for the motion. In this article, we shall explore the three equations of kinematics of circular motion. 

Table of contents

• Definition of uniform and non uniform circular motion
• Average angular velocity 
• Instantaneous angular velocity
• Average angular acceleration
• Instantaneous angular acceleration
• Equations of motion in angular kinematics 
• Practice problems
• FAQs

Definition of uniform and non uniform circular motion 

A type of circular motion in which the speed of the rotating body remains constant throughout is known as uniform circular motion. And a type of circular motion in which the speed of the rotating body does not remain constant throughout is known as non uniform circular motion

In the above figure, represents linear velocity which is tangential to the circle of radius and indicates the angular velocity which is perpendicular to the plane of the circular motion. By definition of circular motion, the magnitude of remains constant but its direction changes.

Average angular velocity 

The total angular displacement divided by the time interval is defined as the average angular velocity . Mathematically,

Instantaneous angular velocity

The instantaneous angular velocity() is defined as the angular velocity at a certain instant of time. 

angular displacement

time

The linear velocity and angular velocity are connected by the relation,  

The unit of angular velocity is

Note: 

Angular velocity

Where indicates the frequency and is the time period.

Average angular acceleration

The total change in angular velocity divided by the total time taken is called the average angular acceleration. Let the angular velocity of a body change from to in a time interval  

The average angular acceleration() can be written as 

Instantaneous angular acceleration

The angular acceleration of a body at a certain instant of time is called instantaneous angular acceleration.

The unit of angular acceleration is

Equations of motion in angular kinematics

The three equations governing angular kinematics are as follows:

final angular velocity

time

initial angular velocity

Angular acceleration 

Angular displacement

The above three equations are analogous to equations of linear kinematics

Note: These equations are only valid for constant angular acceleration.

Video explanation

https://www.youtube.com/watch?v=7DN4ZTt3zuY

Practice problems

Q. A wheel rotates with an angular acceleration given by where is the time, and are constants. If the wheel has an initial angular velocity at , write the equations 

for angular velocity as a function of time .

A. Given, angular acceleration and angular velocity at

Q. A wheel rotates with an angular velocity given by where is the time, and are constants. Write the equations for angle turned as a function of time, . 

A.

Q. A particle moving in a circular path changes its angular velocity uniformly from to in seconds. Find the angular acceleration of the particle. 

(a) (b) (c) (d)

A. a

Given, 

Initial angular velocity, , final angular velocity,

Time,

Q.A wheel has an angular acceleration of and an initial angular speed of In a time of calculate the total angle in radians through which it has rotated.

(a) (b) (c) (d)

A. b

Given,

The angular displacement,  

FAQs

Q.Give an example of non-uniform circular motion.
A. A ball tied to a rope when given a velocity at the lowest point to complete a vertical circle is an example of non uniform circular motion.

Q.What keeps an object moving in a circular path?
A. Centripetal force is responsible for keeping the body moving in a circular path.

Q.Is circular motion possible without force?
A. No, circular motion is not possible without force. Force is needed to change the direction of motion.

Q. What remains constant in a uniform circular motion?
A. The speed remains constant in a uniform circular motion.