Circular Motion: Definition, Formula, and Examples - Coding

Circular Motion is defined as the movement of an object rotating along a circular path. Objects in a circular motion can be performing either uniform or non-uniform circular motion. Motion of a car on a bank road, the motion of a bike well of death, etc. are examples of circular motion.

In this article, we will learn about circular motion and some related concepts, such as examples, equations, applications, etc.

Table of Content

What is Circular Motion?
Equations for Circular Motion
Centripetal Force
Centrifugal Force
Types of Circular Motion
Circular Motion and Rotational Motion
Circular Motion Formulas

What is Circular Motion?

When any object moves in a circular path, it is said to be in circular motion. For example, when a car moves around a circular track at some speed, when the hands of the analog clock are moving, or when you swing an object tied with a rope in a circular path. In all the examples of circular motion, you will generally encounter an object that will be moving at some speed (constant or varying) around a circular path. Generally, the objects tend to move in a straight motion when you apply some force. But for an object to move in a circular motion, there must be some force acting on the object continuously which will turn the object towards the center of the circular path so that the object changes its direction continuously and thus forms a circular motion.

Circular Motion Definition

In physics, circular motion can be defined as the motion in which an object moves when it follows a circular path. In circular motion the object generally moves around a fixed point and the distance between the object and the fixed point is generally fixed.

Circular Motion Example

There are many examples of circular motion which you have already encountered in your life.

Example 1: When you watch the hands of the analog clock moving in a clockwise direction, you will find that the hands are already moving in circular motion. Although it might look like the hands are actually rotating but when you will trace the motion of the tip of the hands, you will find that they are in circular motion. (We will further discuss the difference between rotational motion and circular motion in this article.)

Example 2: When you tie a stone with a rope and swung it in a circular motion (Kindly check your environment before doing this experiment as this may hurt someone).

Example 3: You must have heard that the electrons orbits in a circular path around a fixed nucleus in Bohr’s Atomic model. This also constitutes the example of circular motion as the electrons are going along a circular path (in Bohr’s Atomic model).

Circular Motion in Real Life

There are also situations where you have encountered circular motion of an object in real life too.

Example 1: When we go to some fair (carnival) and we want to ride the Ferris wheel. The pods of the Ferris wheel moves in a circular motion.

Example 2: A car turning on a circular track is also an example of circular motion. Generally these curves are made at an angle known as banked road which helps to reduce the role of friction to the circular motion of the car (Don’t worry if you are not able to visualize this contribution of friction, we will help you visualize this later in this article).

Example 3: In your home when your mother bakes the cake, she used to make the batter for the cake first. For making the batter for the cake she stirs the spoon in a circular motion, where the mixing spoon is follows a circular path equidistant from a central point.

These real life examples helps us visualize the circular motion and we tend to conceptualize the concepts even better.

There are various new terms which comes into the picture when we talk about circular motion. These new terms generally arises from the fact that in circular motion there is angle involved so the terms, such as, angular displacement, angular velocity, or angular acceleration. The image added below shows the Angular Velocity and others in an object performing circular motion.

Uniform-Circular-Motion

Let’s see these concepts one by one.

Angular Displacement

Angular displacement can be defined as the measurement of the amount of rotation an object has gone through in a circular path. In circular motion, the object moves in a circular path, and angular displacement helps us observe the position of the object in a circular path. It can also be understood as the angle made by the position vector of the object between its final and initial position in the circular path.

Angular displacement is a vector quantity. The SI unit of the angular displacement is radians. It is conventionally denoted as θ. The mathematical representation of angular displacement is,

Angular Displacement(θ) = Arc Length/Radius

Therefore,

θ = S/R

where,

S is Linear Displacement Done by Object on Circular Path
R is Distance of Object from a Fixed Central Point (Called Radius)

Angular Velocity

Angular velocity can be defined as the rate of change of angular displacement. It is analogous to the linear velocity as it is the rate of change of linear displacement. Angular velocity can also be understood as the rate at which an object moves in a circular path.

Angular velocity is a vector quantity. It is denoted by ω.

SI unit of angular velocity is radian per second (rad s^-1).

Mathematically, angular velocity can be represented as,

Angular Velocity (ω) = dθ/dt

We know from above that, θ = S/R

Using Angular Displacement in above equation,

ω = d/dt.(S/R)

which gives,

ω = 1/R.(dS/dt)

Finally,

ω = v/R

where,

V is Linear Velocity and V = dS/dt
R is Distance of Object from a Fixed Central Point

Angular Acceleration

Angular acceleration can be defined as the rate of change of angular velocity. It can be understood as the measurement of how fast or slow the angular velocity of an object is changing on the circular path. When any object starts from rest and acquires motion in circular path, it is said to have angular acceleration working on it. For example, when the Ferris wheel starts from the rest and acquires the speed, then the pods of the Ferris wheel gains angular acceleration. When the angular velocity increases, then the angular acceleration is positive. But when the angular velocity decreases, the angular acceleration is negative, i.e., angular deceleration.

Angular acceleration is vector quantity. It is denoted by α.

SI unit of angular acceleration is radian per second squared (rad s^-2).

Angular acceleration can be represented as,

Angular acceleration (α) = dω/dt

We can substitute ω = v/R in above equation to get,

α = d/dt (v/R)

α = 1/R (dv/dt) = 1/R.(a)

Since, rate of change of linear velocity is called linear acceleration, therefore, above equation can be written as,

α = a/R

where,

a is Linear Acceleration of Object
R is Distance of Object from a Fixed Central Point

Equations for Circular Motion

Acceleration which an object faces in circular motion generally has two components:-

Tangential Acceleration (a_T)
Radial Acceleration (a_R)

When the object moves in a circular path, it experiences two different acceleration which works for two different purposes. One acceleration provides the magnitude of acceleration to the object while the other one is responsible for its direction. The acceleration which is responsible for the magnitude is called as tangential acceleration or linear acceleration. The acceleration which is responsible for the direction of the object moving in a circular path is called radial acceleration or centripetal acceleration. Both the tangential and the centripetal acceleration are perpendicular to each other.

Centripetal acceleration acts towards the centre of the circle and keeps the object in a circular path. This centripetal acceleration is further responsible for the Centripetal force. The normal reaction of this force is Centrifugal force which is equal in magnitude and opposite in direction to the centripetal force.

As we know the centripetal acceleration is given as, a_c = V²/R. Since this acceleration is responsible for the centripetal force , therefore the centripetal force is given by,

F = ma_c= mV²/R

We also know from above that ω = V/R

Putting the value of V from above in centripetal force, we get,

F = mRω²

Since the object is moving in a circular path, the object must have taken some time to complete one full revolution. As we know that the time taken by the object to complete one full revolution is defined as its time period. It is denoted by T. A similar but slight different concept is frequency, which is the number of revolution made by the object in one second. Frequency is denoted by ν.

ν = 1/T

In a complete revolution, the object will move a distance of S = 2πR. Therefore, we will have V = 2πR/T

In terms of frequency we can write V = 2πRν. The angular velocity can be written as, ω=2πν. The centripetal acceleration can be written as, a_c = 4π²ν²R

Centripetal Force

Centripetal force is the force which causes any object to undergo in the circular motion. This force is actually responsible for the motion of the object in the circular path.

Centripetal force acts inward towards the center of the circular path. If the centripetal force would be absent, the object will continue to move in the straight path. For example, when we swing a ball tied to string, the ball will start moving in the circular motion.

Centripetal force will be acting (along the string) on the ball will keep the ball in the circular path. The moment you will release the string, the ball will loose its centripetal component and then it will follow a straight path.

Centripetal force F_C is given by, F_C = mV²/R, where m is the mass of the object having linear velocity V and moving along a circular path of radius R. When the time period of the object to complete one revolution is T, then the linear velocity V is given by, V = 2πR/T.

Centripetal force in terms of time period T is given by, F_C = 4mπ²R/T².

Centrifugal Force

Now lets talk about Centrifugal force. Have you been in a car when it is moving on a curve or circular path. You must have feel an outward sensation when the car moves in circular motion. While the car is moving in the circular path due to the centripetal force, what you were experiencing was actually the centrifugal force.

Centrifugal force is actually a pseudo force (not a real force) which is experienced by the object when it is moving in a circular motion. It acts in the outward direction of the circular motion.

Centrifugal force is observed in the non-inertial frame of reference. The magnitude of the centrifugal force is equal and opposite in direction to the centripetal force. That’s why it is considered as the normal reaction of the the centripetal force.

Centrifugal force in terms of terms of linear velocity is given by,

F_C = mV²/R

where,

m is Mass of Object
R is Radius of Circular Path
V is Linear Velocity of Object Moving in Circular Motion

Applications of Centripetal Force and Centrifugal Force

Applications of centripetal force and centrifugal force are observed in our daily lives are,

Centripetal Force

Car Turnings: When the car turns on a curved circular path, the frictional force between the road and the tyre of the car provides sufficient centripetal force to keep the car in the circular path.

Satellites Orbiting the Planet: Gravitational pull of the planet helps providing the centripetal force which is required by the satellite to keep orbiting in a circular stable path.

Particle Accelerator: Due to the magnetic field, the charged particles experiences the centripetal force which keeps them in the circular path.

Amusement Ride: Rotation of the amusement ride such as merry-go-round provides the necessary centripetal force which keeps the rides in circular motion.

Washing Machine Spin Cycle: The clothes in the washing machine are cleaned using the centripetal force which is provided by the rotating drum inside the washing machine.

Centripetal Force Examples

Centrifugal Force

Car Turning: When the car turns on a circular path, the passengers sitting inside the car feels an outward push which is known as centrifugal force experienced by the passengers.

Artificial Gravity Simulator: Artificial gravity simulator rotates simulating an artificial gravity like situation for the rider due to the centrifugal force experience by the rider in the outward direction.

Cloth Dryers: Rotating motion of the dryer forces the cloth to move in circular motion providing centrifugal force to the wet clothes, and forcing the water droplets to move outward from the wet clothes and thus helping the clothes to dry.

Child on a Merry-Go-Round Ride: Child experiences the outward force in a merry-go-round ride, which acts due to the centrifugal force acting on the child due to rotating motion of the ride.

Equatorial Bulge of Earth: You may heard that the Earth is bulged at the equator and flattened at the poles. This is due to the centrifugal force acting on the Earth due its rotation which creates an outward force at the equator.

Types of Circular Motion

Circular motion can be classified into various types based on various factors. In context of physics and mechanics the object can be in circular motion in many different situations. Some common types includes:-

Uniform Circular Motion
Non-Uniform Circular Motion
Planetary Motion
Rotational Motion

But here we will learn only about, two types of circular motion, i.e., uniform circular motion and non-uniform circular motion.

Uniform Circular Motion

When any object moves in a circular path at some constant speed then we say that the object is in uniform circular motion. Their is no change in the speed of the object and hence there is no acceleration produced. However, it is to be noted that the object is moving in a circular direction and the direction of the object is changing at every point of the path. Hence the centripetal acceleration is applying on the object at every point. This acceleration is inward in direction. But the tangential or linear acceleration is zero as the linear velocity is same.

Let’s look at some examples now.

Uniform Circular Motion Examples

Uniform circular motion has many examples which can be seen around us in everyday life. In these examples the speed is constant which results in stable angular velocity. Such examples are discussed below:

Motion of Planets: The planets are revolving around the Sun shows nearly uniform motion. Although the orbital motion are slightly elliptical in nature, they can be considered to be a circular motion.

Ceiling Fan Blades: The blades of the ceiling fan when in full speed shows uniform circular motion, where the blades are moving at a constant speed in the circular path.

Clock Hands: The hands of the clock moves at a contact speed in a particular circular direction. All the three different hands of the clock, i.e., the second hand, the minute hand, and the hour hand moves at constant speed, respectively.

Ferris Ball: The pods of the Ferris wheel when moving at a constant speed, shows a uniform circular motion.

Merry-Go-Round: The merry-go-round is a classic example of uniform circular motion. When the ride starts moving at a constant speed, the carts or the horses of the merry-go-round displays uniform circular motion as it moves along with the circular platform on which it is mounted.

Acceleration in Uniform Circular Motion

Centripetal Acceleration can be written in terms of linear velocity of the object and the radius of the circular path, and is given by,

a_c = V²/R

where,

a_c is Centripetal Acceleration
V is Magnitude of Linear Velocity of Object
R is Radius of Circular Path

Centripetal acceleration is inversely proportional to the radius of the circular path, which means as the radius of the path decreases the centripetal acceleration increases and vice versa.

Non-Uniform Circular Motion

When the object moves in the circular path along a fixed central point at changing speed, more specifically when the magnitude of the velocity changes over a particular period of time, then the object is said to be in non-uniform circular motion. The change in velocity can have implication on radius in non-uniform circular motion as change in velocity can bring the change in the radius of the system of the circular path on which the object is moving.

Let’s look at some examples now.

Examples of Non-Uniform Circular Motion

Examples of non-uniform circular motion shows change in speed, and angular velocity which results in varied motion of the object along the circular path. These examples show changing velocity over time, either increasing or decreasing. Such examples include,

Roller Coster: Suppose a roller coster makes a loop like ride which is circular in motion. Then the roller coster will approach the loop with a particular speed and then in the middle of the loop the speed will decrease but as soon as the coster will approach the end of the loop its speed will increase. Thus showing varying speed profile and therefore, can be an example of non-uniform circular motion.

Car on curved road: In practical life, the car making a turn at a curved road usually lower its speed and thus resulting in change in speed further resulting in negative acceleration, i.e., deceleration. This is also an example of non-uniform circular motion.

Amusement Ride: Small Ferris wheel in amusement park or fair is usually moved with the help of a human, which moves the Ferris wheel at varying speed. Thus the speed of the pods of the Ferris wheel experiences non-uniform circular motion.

Acceleration in Non-Uniform Circular Motion

In non-uniform circular motion, as the speed or the angular velocity of the object changes, it experiences both the acceleration. Therefore, the acceleration in non-uniform circular motion has two components, i.e., the tangential acceleration as well as the centripetal acceleration.

Tangential Acceleration: This component of acceleration acts along the tangent to the circular path. Tangential acceleration generally changes the magnitude of the linear velocity. Tangential acceleration is given by,

a_t = dV/dt

where

a_t is Tangential Acceleration
dV is Change in Velocity
dT is Change in Time

Centripetal Acceleration: This component of acceleration acts towards the centre of the circular path. It is directed inward towards the centre of the circular path. It is also called the radial acceleration. Centripetal acceleration is given by,

a_c = V²/R

where

a_t is Centripetal Acceleration
V is Velocity
R is Radius of Circular Path

Application of Circular Motion

Application of circular motion can be found in our everyday life as well as in practical situations. Various application includes,

Turning of Vehicles at Banked Road

As we know that the centripetal force which is directed towards the centre of a vehicle moving at a circular road is given by, F_C = mV²/R. This force is provided by the force of friction which is between the tyre of the vehicle and the surface of the road.

Note that it is the static friction which provides the centripetal force. Suppose the static frictional coefficient is μ_S and R is the radius of the circular road. Then the maximum speed at which the vehicle can drive at the circular flat road is given by,

V_max = √(μ_SRg)

As the velocity of the vehicle is dependent on the μ_s and R, and the radius is generally not changeable. We can reduce the contribution of the static friction on the vehicle in a circular motion if we can make the road banked at an angle.

Let the angle at which the road is banked be θ.

As the vertical component has no acceleration part, its net force must be zero, therefore,

N cosθ = mg + Fsinθ

Horizontal component is solely responsible for the centripetal force acting on the vehicle, therefore,

Nsinθ + Fcosθ = mv²/R

Since F is less than μ_sN. Therefore, in order to have the maximum velocity, we have to put,

F_S = μ_sN

Therefore, the equation of the vertical components will become,

N cosθ = mg + μ_sNsinθ

And Horizontal component will become,

Nsinθ + μ_sNcosθ = mv²/R

From vertical component, we can get the value of N to be,

$N = \frac{mg}{cos\theta-\mu_ssin\theta}$

Putting this value of N in horizontal component, we get,

$\frac{mg(sin\theta+\mu_scos\theta)}{cos\theta-\mu_ssin\theta} = \frac{mV_{max}^2}{R}$

Rearranging the term and dividing the left hand side of the equation with cosθ, we get V_max to be,

$V_{max} = \sqrt{\frac{Rg(\mu_s+tan\theta)}{1-\mu_stan\theta}}$

Comparing this maximum speed with the maximum speed of the vehicle at the flat road, we can clearly see that this term has some other part in the equation which increases the maximum speed with which the vehicle can move at a banked road in a circular motion.

Differences Between Uniform and Non-Uniform Circular Motion

Below is listed a table to differentiate between uniform circular motion and non-uniform circular motion.

Uniform Circular Motion Vs Non-Uniform Circular Motion
Characteristics	Uniform Circular Motion	Non-uniform Circular Motion
Linear Velocity	Constant magnitude	Varying magnitude (increase or decrease)
Angular Velocity	Constant over time	Can increase, decrease or remain constant over time
Angular Acceleration	Remains Zero as there is no change in angular velocity	Can have varying value as there is change in angular velocity
Centripetal Acceleration	Constant magnitude	Magnitude can change (increase or decrease)
Tangential Acceleration	Zero as the linear velocity is not changing	Non-zero as there is change in linear velocity over time.
Time Period	Constant	Can be constant or changing
Frequency	Constant	Can be constant or changing
Example	Motion of Planets around Sun in the orbit	Car turning on the circular path with varying speed

Circular Motion and Rotational Motion

Below table highlights the difference between the circular motion and the rotational motion.

Difference Between Circular Motion and Rotational Motion
Characteristics	Circular Motion	Rotational Motion
Definition	Motion of object when it moves in circular path	Motion of object when it rotates around an axis
Defined Path of movement	Circular as the object moves along the circular path	Circular but the object rotates along a fixed axis of rotation
Axis of Rotation	Outside the body	Inside the body
Reference Point	Centre of the circular path	Axis of the rotation
Angular Displacement	Angle through which the object has moved along the circular path	Angle through which the object has rotated along the axis of rotation
Angular Velocity	Rate of change of angular displacement with respect to time	Rate of change of rotational speed with respect to time
Angular Acceleration	Rate of change of angular velocity with respect to time	Rate of change of rotational speed with respect to time
Example	Earth revolving around the Sun in an orbit causing seasonal changes	Earth rotating on its own axis of rotation in the orbit causing day and night

Circular Motion Formulas

Circular Motion formulas are added in the table below,

Physical Quantity	Denoted by	Formula
Angular Displacement	θ	θ = S/R
Angular Velocity	ω	ω = V/R
Angular Acceleration (Tangential)	α	α = a/R
Centripetal Acceleration	a_c	a_c = V²/R
Centripetal Force (in terms of V)	F_c	F_c = mV²/R
Centripetal Force (in terms of ω)	F_c	F_c = mRω²
Frequency (in terms of Time period)	ν	ν = 1/T
Linear velocity (in terms of Frequency)	V	V = 2πRν
Angular velocity (in terms of Frequency)	ω	ω = 2πν
Centripetal Acceleration (in terms of ν)	a_c	a_c = 4π²ν²R

Learn More,

Distance and Displacement
Velocity
Acceleration

Circular Motion – Solved Examples

Example 1: Find the angular velocity of the boy who is riding the bicycle at a speed of 10 ms^-1on a circular path of radius 25 m.

Solution:

We have given,

Linear speed of the boy as he is riding the bicycle, V = 10 ms^-1
Radius of the circular path, R = 25 m
We know that, the angular velocity of the boy riding the bicycle on a circular path can be obtained by using the given formula,

$\omega = \frac{V}{R}$

Substituting the value of V and R in the formula of angular velocity, we get,

$\omega = \frac{10\,ms^{-1}}{25\,m}$

$\omega = 0.4\,rad\,s^{-1}$

Therefore, the angular velocity (ω) of the boy who is riding the bicycle at a speed of 10 ms^-1 on a circular path of radius 25 m is 0.4 rad s^-1 .

Example 2: In the above problem, if the mass of the boy is 35 kg then calculate the centripetal acceleration of the boy and also find the centrifugal force acting on the boy.

Solution:

We know that centripetal acceleration is given by,

$a_c = \frac{V^2}{R}$

Substituting the value of V and R in above formula, we get,

$a_c = \frac{{(10\,ms^{-1})}^2}{25\,m}$

a_c = 4 ms^-2

Now, the centrifugal force can be given by the,

$F_c = \frac{mV^2}{R}$

As the mass of the boy is given to be m = 35 kg, therefore,

$F_c = \frac{35\,kg.{(10ms^{-1})}^2}{25\,m}$

F_c= 140 N

Therefore, centripetal acceleration of the boy is 4 ms^-2 and the centrifugal force experienced by the boy is 140 N.

Example 3: Suppose a motorcyclist is making a turn at a speed of 10 ms^-2. How will the force acting towards the centre will change if he doubles its speed?

Solution:

As we know, the centripetal force is given by,

$F_c = \frac{mV^2}{R}$

Since, the centripetal force is directly proportional to the square of the speed, i.e.,

$F_c \propto V^2$

Therefore, when the speed will get doubles, the centripetal force acting on the motorcyclist will increase to 4 times.

Example 4: A car is going in a non-uniform motion on curve of circular path, and its tangential acceleration is given as 3 ms^-2, while its centripetal acceleration is given as 4 ms^-2. Calculate its total acceleration.

Solution:

Given,

Tangential Acceleration (a_t) = 3 ms^-2
Centripetal Acceleration (a_c) = 4 ms^-2
We know that the total acceleration is given by,

$a = \sqrt{a_t^2\,+\,a_c^2}$

Substituting values, we get,

$a = \sqrt{{(3\,ms^{-2})}^2\,+\,{(4\,ms^{-2})}^2}$

$a = \sqrt{9\,+\,16} = \sqrt{25} = 5\,ms^{-2}$

Therefore the total acceleration is 5 ms^-2.

Example 5: An insect is trapped in a circular groove of radius 10 cm moves along the groove steadily and completes 5 revolutions in 100 seconds. What is the angular speed and linear speed of the motion?

Solution:

Given,

Radius, R = 10 cm
Total revolution made = $2\pi \times5$
Time taken, T = 100 s
Therefore angular speed, ω is given by,

$\omega = \frac{Total\,revolution\,made}{Time\,taken}$

$\omega = \frac{2\pi\times5}{100\,s}$

$\omega = 0.31\,rad\,s^{-1}$

The linear speed, V is given by,

V = ωR

$V = 0.31\,s^{-1}\,\times\,10\,cm = 3.1\,cm\,s^{-1}$

Therefore, the angular speed is $0.31\,rad\,s^{-1}$ and the linear speed is $3.1\,cm\,s^{-1}$ .

Circular Motion – Numericals

Q1: A boy with a mass of 25 kg is rides his cycle in a circular path at a speed of 2.5 ms^-1. If the radius of the circular path is 2.5 m, then calculate the centripetal acceleration of the boy.

Q2: A boy is riding a merry-go-round. If he completes 10 revolutions in 20 seconds. Calculate its angular velocity.

Q3: A car of mass 500 kg is moving on a circular path of radius 150 m. The car is moving at a constant speed of 50 ms^-1. How much force is required to keep the car in circular motion?

Circular Motion-FAQs

1. What is Called Circular Motion?

When any object moves in circular path at constant or varying speed, then the motion of the object is said to be in circular motion.

2. What is Uniform Circular Motion and Non-Uniform Circular Motion?

When the object is moving at a constant speed on the circular path, then it is said to be moving in uniform circular motion. Whereas if the object is moving with some changing speed, then the object is said to be moving in non-uniform circular motion.

3. What is Relation Between Linear Speed and Angular Speed?

Linear speed and the angular speed are related by the formula, V = ωR

4. What is Relation between Linear Acceleration and Angular Acceleration?

Linear Acceleration and Angular Acceleration are related by the formula, a = αR

5. What is Centripetal Force and Centrifugal Force?

Centripetal force is an inward force which acts on the object moving along a circular path in order to keep the object in circular motion. Whereas centrifugal force is the outward force which is feel by the object when it moves along a circular path at some speed in a circular motion.

6. What are Tangential Acceleration (a_t) and Centripetal Acceleration (a_c)?

Tangential Acceleration (a_t) is brings changes in the magnitude of the linear velocity, while centripetal acceleration (a_c) brings changes in the direction of the object moving in circular motion. It is directed inward towards the center of the circular path.

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