**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
. The centripetal and the centrifugal forces enable the object to move in the orbit.**centrifugal force**

**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/s^{2}

But V=rω

Then a=rω2

### 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

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