WEEK 9: Circular Motion

Meaning of Circular Motion

Circular motion is the motion of a body around a circle. The simplest form of circular motion is the uniform circular motion, where the speed is constant but the direction is changing.

Consider a body moving in a circular path center O with a constant speed.

The direction at different points are not the same i.e. the direction at A is different from the direction at B. This leads to a change in velocity.
This difference in velocity produces an acceleration directed towards the center of the circle. This acceleration is called centripetal acceleration.
Since there is an acceleration, there is a force directed towards the center of the circle called centripetal force.
In addition to the centripetal force, there is an equal but opposite force which acts outwards from the center of the circle. This force is called the centrifugal force. The centripetal and the centrifugal forces enable the object to move in the orbit.

Definition of Terms Used in Circular Motion

1. Angular velocity (ω):

The ratio of the angle turned through to the elapsed time.

ω= Angular velocity

ω=angular displacementtime=θt

The S.l unit is rad/sec

2. Tangential velocity (V):

This is the linear velocity whose direction is along the tangent to the circumference of the circle.

V=displacement(s)time(t)=st=rθt

But ω=θt

Then V=rω

The unit is m/s

3. Centripetal acceleration (a):

This can be defined as the acceleration of a body in uniform circular motion whose direction is towards the centre of the circle. It is given as:

a=V2r

The unit is m/s²

But V=rω

Then a=rω2

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4. Centripetal force (F):

It is defined as that inward force that is always directed towards the centre of the circle required to keep an object moving with a constant speed in a circular path.

Centripetal force =mass×centripetal acceleration

F=mv2ror

F=rω2=ωVr=ma

The unit is Newton

5. Centrifugal force:

This force is equal in magnitude to the centripetal force but opposite in direction. (it is always directed away from the centre of the circle)

F=−mv2rorF=−rω2

6. Period (T):

This is the time taken for a body to complete one revolution round the circle.

Displacement = 2

Time = T

Velocity = v

v=displacementtime=2πrTT=2πrv

7. Frequency (f):

It is the number of revolutions in one second.

f=1TT=v2πr

The unit is Hertz or per seconds. (i.e Hz or s^-1)

Calculations on Circular Motion

Question 1:

A stone of mass 2kg is attached to the end of an inelastic string and whirled round two times in a horizontal circular path of radius 3m in 3 sec, find:

(i) Angular velocity

(ii) Linear velocity

(iii) Centripetal acceleration

(iv) Centripetal force

(v) Centrifugal force

Solution

(i) ω=angular displacementtime=θt

Where is the angular displacement and ω is the angular velocity

θ=360×2=720o (ie two times)

π=180oθ=4πradω=4π3=1.33πrad/sec

(ii) v=rω=3×1.33π=3.99πm/s

(iii) a=v2ra=(3.99π)23a=5.31π2m/s2

(iv) F=ma=2×5.31π2=10.62π2N

(v) F=−mv2r=−10.62π2N

GENERAL EVALUATION

Explain the following terms (i) Angular velocity (ii) Tangential velocity (iii) Centripetal acceleration
A body of mass 10kg is attached to the end of an inelastic thread and whirled round in a circular path of radius 0.3m, if the body makes a complete revolution in 3 sec find

(i) angular velocity

(ii) linear velocity

(iii) centripetal acceleration

(iv) centripetal force

(v) centrifugal force