# Circular Motion Notes With JEE NEET MCQS Physics Class 11 CBSE Study Material Full Chapter Download pdf-Anand Classes

# What Is Uniform Circular Motion?

The movement of a body following a circular path is called a circular motion. Now, the motion of a body moving with constant speed along a circular path is called Uniform Circular Motion. Here, the speed is constant but the velocity changes.

If a particle is moving in a circle, it must have some acceleration acting towards the centre which is making it move around the centre. Since this acceleration is perpendicular to the velocity of a particle at every instant, it is only changing the direction of velocity and not magnitude and that’s why the motion is uniform circular motion. We call this acceleration centripetal acceleration (or radial acceleration), and the force acting towards the centre is called centripetal force.

In the case of uniform circular motion, the acceleration is:

a_{r} = v^{2}/r = ω^{2}r

If the mass of the particle is m, we can say from the second law of motion that:

F = ma

mv^{2}/r= mω^{2}r

This is not a special force, actually force like tension or friction may be a cause of origination of centripetal force. When the vehicles turn on the roads, it is the frictional force between tyres and ground that provides the required centripetal force for turning.

# What Is Circular Motion?

**Circular motion** is described as a movement of an object while rotating along a circular path. Circular motion can be either uniform or non-uniform. During uniform circular motion, the angular rate of rotation and speed will be constant, while during non-uniform motion the rate of rotation keeps changing.

Some of the most common examples of circular motion include man-made satellite that revolves around the earth, a rotating ceiling fan, a moving car’s wheel, the blades in a windmill, and gears in gas turbines.

A particle is said to execute circular motion when it moves along the circumference of a circular path. An important aspect of circular motion is that the direction of motion is changing continuously unlike in the case of linear motion. Hence, circular motion can also be described in terms of angular variables.

# Angular Variables

**Angular Displacement**

It is defined as the angle turned by a rotating particle per unit time. It is represented by ∆θ and measured in radians.

**Angular Velocity**

It is defined as the rate of change in angular displacement of a particle in a circular motion. It is denoted by

**ω** = lim_{∆t→0} (∆θ/∆t) = dθ/dt

Angular velocity is measured in rad/s. Apart from angular velocity and angular speed, a particle in circular motion also possesses linear velocity and corresponding linear speed.

**v**= d**s**/dt

v= |d**s/**dt|; **s** is the displacement of the particle

# Relation between Linear Speed(V) and Angular Speed(Ω)

In vector form

**v= ω x r**

Where **r **is the position vector of the particle measured with respect to the centre of the circle.

(Or)

v = rω

The acceleration of a particle in circular motion has two components:

- Tangential acceleration a
_{t}: This is the component of acceleration in the direction of the velocity of the particle.

a_{t}= d|**v**|/dt

- Radial acceleration a
_{r}: This is the component of acceleration directed towards the centre of the circle. This component causes a change in the direction of the velocity of a particle in a circular motion.

a_{r}= v^{2}/r = rω^{2}

**Angular Acceleration**

It is defined as the rate of change of angular velocity of the rotating particle. It is measured in rad/s^{2}

α = dω/dt = d^{2}θ/dt^{2}

Circular motion can be uniform and non-uniform depending on the nature of acceleration of the particle. The motion is called uniform circular motion when the particle is moving along a circular path possessing a constant speed.

During circular motion, the velocity vector changes its direction at each point on the circle. This implies that the radial component of acceleration is always non-zero. The tangential component can take a positive or negative value in the case of non-uniform circular motion and a zero value in the case of uniform circular motion.

We understand that the acceleration of a particle in a circular motion is always directed towards the centre and is given by v^{2}/r. Applying Newton’s second law of motion in this situation;

F_{cen }= mv^{2}/r

Where m is the mass of the particle

F_{c }is the centripetal force directed towards the centre of the circular path.

Examples of centripetal forces include gravitational force, the tension in a string and friction.

## NOTE

So if a particle is moving in a uniform circular motion:

1) Its speed is constant

2) Velocity is changing at every instant

3) There is no tangential acceleration

4) Radial (centripetal) acceleration = ω2r

5) v=ωr

In case of non-uniform circular motion, there is some tangential acceleration due to which the speed of the particle increases or decreases. The resultant acceleration is the vector sum of radial acceleration and tangential acceleration.

# Uniform Circular Motion Examples

Following are the examples of uniform circular motion:

- Motion of artificial satellites around the earth is an example of uniform circular motion. The gravitational force from the earth makes the satellites stay in the circular orbit around the earth.
- The motion of electrons around its nucleus.
- The motion of blades of the windmills.
- The tip of second’s hand of a watch with a circular dial shows uniform circular motion.

# Application of Newton’s Laws of Motion

Let us understand the application of Newton’s laws of motion in two different scenarios. The motion of a vehicle will be on either:

### At Level Road

At an instant when a vehicle makes a turn on a level road, there are three different forces acting.

- The weight of the car W= mg
- Frictional force f
_{f} - Normal reaction N

In the vertical direction N-mg = 0

N = mg

The centripetal acceleration for circular motion is provided by static friction. This friction holds back the vehicle from moving away from the circle.

f_{f} = mv^{2}/r ≤ µN

f_{f} = mv^{2}/r ≤ µ mg

µ is the coefficient of static friction

_{max}= √µ rg

This implies that for a circular path of given radius and µ, there is a maximum possible speed of the vehicle.

# Solved Problems

**Question 1:** What is the angular velocity of a particle in a circular orbit, rotating at the rate of 200 times per minute?

**Solution:** In the given problem, the frequency of rotation is 200 times per minute

ω= 2πf

= 2π x 200/60 = 20.93 rad/s

**Question 2:** An electrically powered wheel is rotating about its axis at the rate of 900 rotations per minute. When the power is turned off it comes it rest in 1 minute. Find the angular deceleration of the wheel in rad/s^{2}.

**Solution:** From the laws of motion in the case of rotational motion, We know that ω = ω_{0 }+ αt.

From the given data :

ω_{0 }= 900rpm = 900 x 2π/60 rad/s

ω = 0 and t= 60s

Hence, angular deceleration α = π/2

**Question 3:** A bus is moving in a circular track of radius 1000cm with a speed of 10m/s. A pendulum is suspended from the roof of the bus by a light rigid rod of length 100cm. Find the angle made by the rod with the track. (Use g=10m/s^{2}).

**Solution:**

tan θ = v^{2}/rg = 10^{2}/10×10

tan θ = 1

tan θ = 45⁰

**Question 4:** The minimum velocity with which a car driver must traverse a curve of radius 150 m and coefficient of friction 0.6 to avoid skidding is? [IIT-JEE 2002].

**Solution:** In the relation mv^{2}/r =µN substitute N = mg

We get v^{2} =µrg

v^{2 }= 0.6x150x10

v= 30m/s

**Question 5:** Which of the following statements is false for a particle moving in a circle with a constant angular speed? [AIEEE 2004]

a)The velocity vector is tangent to the circle

b) The acceleration vector is tangent to the circle

c) The acceleration vector points to the centre of the circle.

d) The velocity and acceleration vectors are perpendicular to each other.

**Solution:** Option b)

# Frequently Asked Questions on Circular Motion

### What type of motion is called circular motion?

The motion of an object in a circular path along the circumference of a circle or rotation along a circular path is called circular motion.

### What are the types of circular motion?

Circular motion can be either uniform or non-uniform.

### Give an example of circular motion.

The movement of the satellite in a circular orbit, ice skaters going in a circle with constant velocity, and cars moving in a circular path are examples of circular motion.

### What is the value of instantaneous acceleration in a uniform circular motion?

The instantaneous acceleration in a uniform circular motion is given by

a = v^{2}/R

v is the velocity of the object.

R is the radius of the circle.

The direction is perpendicular to velocity and directed inwards along the radius.

### Name the device used for measuring the speed of rotation.

The tachometer is a device that is used for measuring the speed of rotation.

### What happens to the velocity vector of a particle, if the particle is moving in a circle, at equal angles in equal times.

The velocity vector of a particle will change its direction. This is because it is always directed in the direction of the tangent to the circle.

### Which property is conserved when a particle is moving with constant angular velocity?

When a particle is moving with constant angular velocity, the energy of the particle is conserved. This is because in a uniform circular motion, kinetic energy remains unchanged and the momentum of the particle varies with a change in velocity.

### What happens to the direction when a vector is multiplied by a positive number?

The work done by the body when it moves along a circle with a constant speed is zero. This means that the work done by the centripetal force is zero.

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