Dynamics of Circular Motion

In Physics, the concept of circular motion has multiple usages. Once understood the concept is applied to solve the problems of rotation. Understanding the importance of the topic we have compiled a separate article explaining the concept of circular motion. 

In this particular article, we shall focus on understanding the dynamics of circular motion. The explanation will focus mainly on the following concepts-

Table of Content - 

• Circular motion - introduction with example

• What is the Dynamics of a circular motion

• Right-hand rule

• Dynamics of a Uniform circular motion 

• Dynamics of a Non-uniform circular motion

• FAQs 

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 has 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

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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 +  at 

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

Key Points from the Chapter -

• Centrifugal and centripetal forces are equal in magnitude and are also opposite in direction.

• Centripetal and centrifugal forces cannot be termed as action and reaction since action and reaction never act on the same body.

• To complete the motion in a vertical circle is the basic requirement for a body under limited conditions.

• For example, the stress on the string must not vanish before it reaches the highest point.If the stress vanishes earlier, it will be devoid of the necessary centripetal force required to keep the body moving in a circle.

• Centrifugal force is known as the fictitious force which acts on a body, rotating with a uniform velocity in a circle and along with the radius away from the center.