Dynamics of Circular Motion

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What is Circular Motion?

A body moving along the circumference of the circle with a constant speed is said to be exhibiting a circular motion.

For example, a car having a circular motion with a speed of  8 m/s along the circumference of 24 meters.

At a uniform speed, it will complete one cycle in 3 seconds.

It means in every circle, around the 24 m circumference of the circle, a body would take the same time of 4 seconds. 

So, this relationship between the circumference of a circle, the time to complete one revolution, and the speed of the body can be described in terms of average speed.

So, Average speed = distance/time = circumference/time = 2 * π * r/T

As circumference  = 2 * π * r

Dynamics of Circular Motion

Consider a body, moving along the circular path of radius r, in a clockwise direction in the plane of a paper.

Let's say the axis of the circular motion is passing through the center O, perpendicular to the plane of a paper.

As you see in Figure.1 below:

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The angle traced from P to Q is called the angular displacement, given by,

                                  Ө = PQ/r = S/r

It is a vector quantity.

Its direction can be given by the right-hand rule.

Right-Hand Rule

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It states that if the fingers are curled in the direction of motion as if they are gripping the axis of rotation. The thumb that is held perpendicular to the curvature of the fingers represents the direction of the angular displacement vector.

As it exhibits a circular motion, it has a velocity too, and that velocity is the angular velocity.

Angular velocity is the rate of change of angular displacement. It is symbolized by ω.

              Where   ω  = v/r

(image to be added soon).

It is a vector quantity. 

By the right-hand rule, the thumb represents the direction of angular velocity.

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For a body having anticlockwise rotation, by the right-hand rule, the direction of ω is along the axis of a circular path and directed upwards, while for clockwise rotation, ω is directed downwards.

Dynamics of Uniform Circular Motion

The natural tendency of the body is to move uniformly in a straight line.

When we apply a centripetal force to it, it is forced to move along the circle.

Let’s consider a body, uniformly moving along the circumference of the circle when a pseudo-force is applied to it.

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The pseudo-force that acts along the radius and is directed towards the center of the circle is called the centripetal force.

According to Newton's first law of motion, the body cannot change its direction of motion, an external force is required to maintain its circular motion.

However, this body continuously changes its direction of motion by itself, and there is a change in the velocity as well, that’s why it undergoes acceleration, called the radial centripetal acceleration.

                      a  = v2/r

We know that F = ma

                       F = mv2/r

Dynamic of Uniform Circular Motion

While moving along the circle, the body has a constant tendency to regain its natural linear path. The tendency gives rise to a centrifugal force.

We can consider the centripetal force as the reaction of the centripetal force.

This means the centrifugal force is always equal and opposite to the centripetal force.

So, centrifugal force = mv2/r, and it acts along the radius, but away from the center of the circle.

The centripetal and the centrifugal forces are the forces of action and reaction, respectively.

Let’s say a stone is tied to one end of the string and the other end is rotated in a circle.

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As you can see in Figure.

When a centripetal force F1 is applied to the stone by the hand. It is pulled outward by centrifugal force, F2 acting on it because it tends to regain its natural linear motion.

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Dynamics of Non-Uniform Circular Motion

Consider a body moving with an angular velocity, ω.                        

It can change either its direction (clockwise or anticlockwise) or change its magnitude, while the axis of rotation remains fixed.

So, the position vector ‘r’  remains constant.

Since v = rω 

Now, differentiating it with respect to time, we get,

         dv/dt = ωdr/dt + rdω/dt

 As a = dvdt,  dr/dt = v, α (angular acceleration) = dω/dt

       = vω + rα

       a = ac +  a

Here, ac = radial or centripetal acceleration, which is the measure of the rate of change of the velocity of the particle in radial direction.

at = tangential acceleration, which is the measure of the rate of change of the magnitude of the velocity of the particle in the tangential direction.

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The magnitude of the resultant acceleration in the circular motion is given by,

                     a  = |a| = √ac^2 + at ^2

FAQ (Frequently Asked Questions)

1. What is the Example of Circular Motion?

Some real-life examples of circular motion are:

  1. An artificial satellite revolving around the earth at a constant height.

  2. The motion of the blades in the windmills.

2. What Is the Example of Non-Circular Motion?

Some real-life examples are:

  1. Oscillation of a pendulum.

  2. A person jogging in the park.

3. What is the Other Name of Non-Uniform Motion?

A body that covers unequal distances in equal intervals of time. It can be called accelerated motion or non-linear motion.

Some examples of non-uniform motion are: 

  1. The motion of a train.

  2. Movement of an asteroid.

4. An Aircraft Executes a Horizontal Loop of Radius 5 Km With a Speed of 900 Km/h. Compare Its Centripetal Acceleration (ac) With the Acceleration Due to Gravity.

Here, r = 5 km = 5000 m 

velocity, v = 900 kmph = 900 x (1000 m)/ (60 x 60s) = 250 m/s

We know that centripetal acceleration, a = v2/r

Putting the values of r and v in the above formula, 

           a = (250)2/5000

On solving, we get,

                  a  = 12.5 ms-2

Now, comparing a to g: (g =9.8ms-2)

 a/g  = 12.5/9.8  = 1.27