CIRCULAR MOTION PHYSICS FORM 5

CIRCULAR MOTION PHYSICS FORM 5

CIRCULAR MOTION Is the motion of the body around the circular track

There are two types of circular motion

(i)  Uniform circular motion

(ii)  Non-uniform circular motion

(i) UNIFORM CIRCULAR MOTION

This refers to the motion of a particle in a circular path that moves with a uniform speed. The word “uniform” refers to the constant speed.

It means that in a uniform circular motion, the object covers equal distances along the circumference in equal intervals of time i.e. speed is constant. Although the magnitude of the velocity (speed) remains constant, the direction of the velocity is changing continuously.

Therefore, the object is undergoing acceleration. This is called centripetal acceleration and is directed radially towards the center of the circle as shown in the figure 1;

Figure .1

(ii) NON UNIFORM CIRCULAR MOTION

This refers to the motion of the particle in circular path moves with a non uniform speed.

The speed of the particle in circular motion is different at different points along the circular path.

RELATIONSHIP BETWEEN LINEAR DISPLACEMENT AND ANGULAR DISPLACEMENT

Consider the particles of mass m moving around the circular track with angular speed ( 

) as illustrated on the figure 2

Figure.2

From the figure above

O    =   Origin of the circle (circular track)

Q    =    Angular Displacement

S    =     Linear Displacement

R   =     Radius of the circular track

=     Angular velocity

Consider the particles moving from Q to P from the figure above

Figure.3
From the figure above

If 

is very small

Therefore

Note

The angle 

in radian is given by

When 

the length of the arc which is the circumference of the circle is 

TERMS RELATING TO CIRCULAR MOTION

Angular Displacement (

)

Is the angle in radian traced at the center of the circle by the body moving around the circular track.

Or

Is the angle turned by an object moving along a circular path in a given time. Consider an object moving along a circular path with  center O as shown in figure 4.

Figure .4

Let us consider O as the origin of our coordinate system.

Suppose initially at t = O, the object is at point P.

At time t₁, the object is at point P₁ and its angular position is 

_1. At time t₂, the object is at point P₂ and its angular position is 

_2. During the time interval 

the angular displacement is 

Angular Velocity ( 

)

Is the angular displacement traced per unit time.

Or

Is the rate of change of angular displacement of a body moving along a circular path

If the angular displacement of an object is 

during the time interval

then the angular velocity of the object is

SI unit of angular displacement is rad/s

The angular velocity of an object moving along the circular path at any instant of time is called instantaneous angular velocity

It is denoted by 

.  It is  given by the limit of 

as 

approaches zero

Angular Acceleration 

Is the rate of change of angular velocity

If change in the angular velocity of an object is 

during time interval

, the average angular velocity is given by

The SI unit of angular acceleration is rad/s²

The angular acceleration of an object moving along a circular path at any instant of time is called instantaneous angular acceleration.

It is denoted by

.  It is given by the limit of 

as 

approaches zero.

∝ =   ^Δω/_Δt

lim Δt → 0

Time Period (T)

Is the time taken by the body moving along a circular path to complete one revolution.

It is denoted by T and its unit is second

For example, if an object completes 120 revolutions in 30 seconds, its time period is given by

It means that the object will complete one cycle in 0.25_s

Frequency (ƒ)

Is the number of revolution completed by the object moving along a circular track in one second.

It is denoted by f and its SI unit is s^–1 or hertz (Hz)

Thus in the above case, the object completes 120 revolutions in 30 seconds. Therefore, the frequency of the object is

It means the object will complete 4 revolutions in one second.

Relation between T and f

Suppose an object executing circular motion has frequency ƒ.

It means that the object completes f revolutions in 1 second.

-Therefore, the time taken to complete one revolution is 1/f.

Relation between 

, f and T

When an object executing circular motion completes one revolution, angular displacement θ and time taken is T.

From

Relation between V and 

As an object moves along the circumference of the circle, it has linear velocity V which is always Tangent to the circular path at every instant as shown in figure 5.

Figure . 5

The relation between the linear velocity V and angular velocity w of the object can be found as under

But

But

This is an important relationship between the circular motion of an object and the linear motion that results from rotation.

Relation between linear acceleration and angular acceleration.

The relation between linear acceleration 

and angular acceleration 

can be found as follows.

EQUATIONS OF UNIFORM MOTION AS APPLIED TO CIRCULAR MOTION.

Consider the particle of mass m moving around the circular track with uniform angular acceleration.

1)  To Derive 

From

At 

2)  To derive 

From    

At 

3)  To derive 

From

When 

Centripetal Acceleration (a_c)

Is the acceleration possessed by the body which is moving around the circular track and always directed towards the center.

It is also called radial acceleration because it always acts radially towards the center of the circle.

This acceleration must be in the same direction as it passes e towards the center of the circle.

For a body which is moving with constant angular velocity ω along a circular path of radius r, the magnitude of the centripetal acceleration to be given by

If the linear speed of the particle is V, then the centripetal acceleration is given by

It’s SI unit is m/s²

For non uniform circular motion, acceleration has two components, centripetal component and tangential component (at).
The magnitude of the resultant acceleration is

To show that the centripetal

Acceleration = 

Consider a particle moving with constant speed V along an arc NOP as in figure 6.

Figure. 6

The x – component of velocity of the particle has the same value at P as at N and therefore its x-component of acceleration a_x is zero

As the particle moves from N to P its y-component of velocity changes by 2Vsinθ.

If this takes place in a time interval t its y – component of acceleration, ay is given by

The speed of the particle along the arc is V, and therefore

Sub equation (ii) into equation (i)

If N and P are now taken to be coincident at O, then 

approaches to zero, and 

has its limiting value of 1 in this case.

Thus at O, a_x = O and a_y = 

and therefore the acceleration is directed along Oz i.e. towards the center of the circle.

Alternately

Figure  7

From the figure 7 above, the distance of displacement S arc is given by

S = r 

Differentiate by product rule

But per body moving in circular path of common radius (Since radius is constant)

Then

Make 

the subject

From

Differentiate by product rule

At,   

Substitute equation (ii) into equation (i)

Divide by t on the 

———————————————————————— 

Since

Also from

When the body is moving around the circular track, velocity remains constant

Substitute equation 

into equation 

Equate the equation 

and equation 

Centripetal Force (F_c)

Is the force possessed by the body moving around circular path and always directs the body towards the centre

OR

Is the force acting on a body moving along a circular path with uniform speed and is directed towards the centre of the circle.

From

If

Also

More over

From

But

No work is done by the centripetal Force

Work done 

F.S Cos 

= 

Some Common Examples of Centripetal Force

(i)  In the case of planets orbiting  around the sun the centripetal force is provided by the gravitational force of attraction between the planets and the sun.

(ii)  In the case of an electron moving around the nucleus of the atom, the centripetal force is provided by the electrostatic force of attraction between the electron and proton.

(iii)  When a particle tied to a string and whirled in a horizontal circle then the tension in the string provides the centripetal  force.

(iv)  When charged particle describing a circular path in a magnetic field, then magnetic force exerted on a charged particle that set up  the centripetal force .

(v)    When a vehicle moves in a circular path on a level road the force of lateral friction between the wheels and the road provides the centripetal force.

Application of Centripetal force in Every Day Life

(a)   In separating honey from bees wax

(b)  In separating cream from milk

(c)   In separating sugar crystals from molasses

(d)  In spin drier machines, water particles fly off tangentially through holes in the wall of the machine.

Centrifugal Force (Fictitious Force)

Is the force which does not really act on a body but appears due to the acceleration of the frame

In order to move a body in a circular path, a centripetal force 

is required.  This force acts along the radius towards the centre of the circle.

The reaction of this centripetal force is the centrifugal force.

Both these forces are equal in magnitude but opposite in direction and act on different bodies.

Force example, consider the case of a stone tied at one end and rotated in a circle as shown below.

Figure 8

Note that the centripetal force F₁ is applied on the stone by the hand and acts towards the center.

The centrifugal force F₂ acts on hand and pulls it away from the center of the circle.

Centrifugal means center – Fleeing or away from the center.

Therefore, the outward force in circular motion is called centrifugal force.

The magnitude of centrifugal force is the same as that of the centripetal force and its direction is opposite to that of the centripetal force.