Non-uniform Circular motion - Definition, Practice problems, FAQs

One day, Nikita goes to the amusement theme park, where she comes across a rollercoaster. She watches the roller coaster go around its track, stop at a certain point, and then speed up before it slows down again. This fascinated her: why didn’t the rollercoaster just maintain the same speed throughout? She glances across the park to see a baby girl revolving a stone tied to a string with her hand; upon looking closely, she notices that the stone went around and around with different speeds. In other words, one can call such examples where the speed of the object is not constant throughout as non-uniform circular motion. The stone is said to undergo a type of non-uniform motion called “vertical circular motion”. In this article, let us explore non-uniform circular motion in detail!

Table of contents

• Definition of non uniform circular motion
• Radial acceleration
• Tangential acceleration
• Net acceleration
• Practice problems
• FAQs

Definition of non uniform circular motion

If the particle in a circular motion does not maintain a uniform speed throughout the motion, then the particle is said to undergo nonuniform circular motion. Roller Coasters, motion of a bob tied to a string, a boy riding a bicycle going around are examples of non uniform circular motion. 

In the following figure, the velocity of the particle keeps changing as the particle keeps traversing the path. There are two components of acceleration : called the centripetal or radial acceleration directed inward and tangential acceleration called directed tangentially. 

Radial acceleration

Consider a particle of mass moving along a circle of radius . The radial or centripetal acceleration is directed inwards. Let and be two different velocities along two sides of the triangle . Let indicate the third side of the triangle. Then according to the vector law of addition,

is the angle between and

For small angles ,

∴

indicates the angular velocity.

Now indicates the linear velocity

.

Tangential acceleration

The component of acceleration directed perpendicular to the radius of the circle( i.e tangential) is called tangential acceleration. This is responsible for changing the speed of the particle.

Now 

, indicates the angular acceleration.

Net acceleration

Let indicate the net acceleration of the particle. Now 

But

Practice problems

Q. A particle begins to move with a tangential acceleration of constant magnitude in a circular path. If it slips when its total acceleration becomes find the angle through which it would have turned before it started to slip?

(a) radian (b) degree (c) radian (d) degree

A. a

Given, ,

centripetal acceleration

tangential acceleration

angular displacement

Centripetal acceleration

radian

Q. A body revolves in a circle of radius Its speed varies according to time given by the relation , where is in and is in seconds.

a. Calculate its tangential acceleration at .
b. Calculate its total acceleration at .

(a) Tangential acceleration 

(b)Tangential acceleration 

( c )Tangential acceleration 

(d)Tangential acceleration 

A. b

Given, speed

Tangential acceleration 

Q. A body moves in a circular path having radius . Its speed varies with time according to the relation given by , where is in second and is in . Find the radial and tangential acceleration at .

(a) (b) (c) (d) zero, zero

A. c

Given, 

Speed,

Radial acceleration

Tangential acceleration 

Q. A car is traveling in a circular curve that has a radius of If its speed is and is increasing uniformly at , determine the magnitude of the acceleration at this instant.

A.

Given, tangential acceleration

Radius and speed

Radial acceleration

Net acceleration

FAQs

Q.Write the difference between uniform and non-uniform circular motion.
A. Non-uniform circular motion involves motion of a particle along a circular path with variable speed. Uniform circular motion involves motion of a particle with constant speed.

Q. What is the work done by the centripetal force ( in non uniform circular motion?
A. In a non uniform circular motion, the centripetal force and displacement are perpendicular to each other. Hence, work done

Q. What is the angle between tangential and radial acceleration?
A. The tangential and radial acceleration are perpendicular to each other, hence angle between them is

Q. Does the radius of a circle affect the tangential acceleration?

A. . It is independent of radius.