Uniform Circular Motion

Uniform Circular Motion as the name suggests, is the motion of a moving object with constant speed in a circular path. As we know, motion in a plane only has two coordinates, either x, and y, y and z, or z and x. Except for Projectile motion, circular motion is also an example of motion in a 2-D plane.

In a uniform circular motion, the object moves with constant speed but not with constant velocity as the direction of the motion is due to the circular path always changing. From the motion of electrons in Bohr’s Atomic model to the motion of the hands of an analog clock, we can see Uniform Circular Motion around us. 

In this article, we will learn about the details of Uniform Circular Motion i.e., formulas related to uniform circular motion, examples, and the equation of motion of the uniform circular motion.

Uniform Circular Motion Definition

Uniform Circular motion is the 2-dimensional motion in which the object moves with a uniform speed in a fixed circular direction but since the direction of the object keeps on changing at each and every point, the velocity keeps on changing as well, and the direction at every point is the direction towards the tangent.

Uniform Circular Motion Examples

There are several examples of Uniform Circular Motion such as, a car driving with constant speed around a circular racetrack, the constant speed motion of the carnival ride marry go round, a ferries wheel rotating at a constant speed, and all the other objects such wheels, rotating ball, vinyl record in record player moving with constant speed, etc.

Terms Related to Circular Motion

Object performing Circular Motion, there are several terms related to it which needed to be defined, such as angular displacement, angular velocity, angular acceleration, centripetal, and centrifugal force, etc. These terms are explained in detail as follows:

Angular Displacement

In the circular path, angular displacement is the difference of angle subtended by the position vector to the center from the initial to the final state. The unit of the angular displacement is Radians. The formula for angular displacement is,

dθ = dS/r

θ = S/r

Where, 

• dS or S is the Linear displacement, and
• r is the radius of the circle.

Note: This formula (θ= S/r) is also the relation between linear and angular displacement.

Angular Velocity

Angular velocity is defined just like linear velocity, that is, Angular velocity is the rate of change of angular displacement. The unit for angular velocity is Radian/sec. It is given by the formula:

ω = dθ/dt 

ω = θ/t

Where, 

• dθ or θ, is the angular displacement of the particle, and
• dt or t is the time taken for the angular displacement.

Angular Acceleration

The angular acceleration is defined as the rate of change of angular velocity (w), The unit for angular acceleration is given as Radian/sec². The linear acceleration is related to angular acceleration as, a_l = αr

The angular acceleration is given as, 

α = dw/dt

OR

a=v²/r

Where, 

• α is angular acceleration, and 
• dw/dt is the rate change of the angular velocity,

Note: The value of angular acceleration is always zero in uniform circular motion since the angular velocity is constant.

Centripetal and Centrifugal Force

When the object travels in a circular motion, at every point, some acceleration is experienced by the object, the acceleration acts towards the center of the circle which makes the object move in that circle. The acceleration is known as Radial acceleration or Centripetal acceleration. 

In a uniform circular motion, the force acting towards the center is called centripetal force and in order to balance that force, the force that acts outside the circle is known as centrifugal force.

The mathematical expression for the Force acting toward the center is given as,

F_c = mv²/r

Note:

• The direction of Velocity is always tangent to the circle at all points.
• The acceleration vector will always be perpendicular to the velocity vector and hence, will always point toward the center.
• The Centripetal force always acts towards the center.

Learn more about, Centripetal and Centrifugal Force

Relation between Linear and Angular velocity

As we know,

[Tex]|v| = r\frac{\Delta \theta }{\Delta t}  [/Tex]  . . .(1)

[Tex]\left[\omega =\frac{\Delta \theta }{\Delta t} \right]        [/Tex]  . . .(2)

Using, (1) and (2) we get,

[Tex]|v| = \frac{\Delta r\theta }{\Delta t} [/Tex]

As r is the radius of the circular path,

[Tex]|v| = r\frac{\Delta \theta }{\Delta t} = r \omega \quad \left[\omega =\frac{\Delta \theta }{\Delta t} \right] [/Tex]

Thus, 

|v| = rω

Hence, it is the required relation between Linear and Angular velocity.

Note: Using this relation, we can find the relation between Linear and Angular Acceleration, i.e., 

The following table shows all the relations, between linear and circular motion,

Circular	Linear	Relation
θ	S	θ = Sr
ω	v	ω = v/r
α	a	α = a/r

Equations of Motion for Uniform Circular Motion

The object moving in a uniform circular motion will have a certain position and speed at each point of time and hence, can be denoted by a position vector. Let us consider that the particle has a position vector [Tex]\vec{r}(t) [/Tex] with an amplitude of P. As circular motion is a 2-dimensional motion, the vector is traveling in both x and y coordinates hence, it will have components of both the x and y-axis. In general, if the object is moving with a uniform angular frequency of ‘w’, at a time ‘t’, it will have the following values,

Position vector (P) ⇢ [Tex] \vec{r}(t) [/Tex]

X-Component of position vector ⇢ [Tex]|\vec{r}(t)|Coswt [/Tex]

Y- Component of position vector ⇢ [Tex]|\vec{r}(t)|sinwt [/Tex]

Where, w is given as, [Tex]w=\frac{2\pi}{T} [/Tex]

T= Time Period

Equation for Position Vector, [Tex]\vec{r}(t)=iP|\vec{r}(t)|coswt+jP|\vec{r}(t)|sinwt [/Tex]

The velocity vector can easily be obtained by differentiating the position vector w.r.t time,

[Tex]\vec{v}(t)=-iwP|\vec{r}(t)|sinwt+jwP|\vec{r}(t)|coswt [/Tex]

Similarly, the acceleration vector can be easily obtained by differentiating the velocity vector w.r.t. time,

[Tex]\vec{a}(t)=-iw^2P|\vec{r}(t)|coswt-jw^2P|\vec{r}(t)|sinwt [/Tex]

From the above equation, the relationship between the acceleration vector and position vector can be easily calculated,

[Tex]\vec{a}(t)=-w^2|\vec{r}(t)| [/Tex]

Read More,

• Centripetal Acceleration
• Motion in a Vertical Circle
• Banking of Road

Sample Questions on Uniform Circular Motion

Question 1: Give Practical examples of uniform circular motion.

Solution:

Practical examples of Uniform Circular motion:

• The motion of the electrons in an atom around the nucleus in Bohr Atomic Model.
• Wall of Death: The bike has a normal force acting towards the centre which makes it move in a circle.
• The Artificial satellite moving around the Earth is an example of uniform circular motion. The gravitational force from the centre of the earth exerted upon the satellite acts as the centripetal force for it.

Question 2: A cyclist is taking a turn with a speed of 5m/sec, if he doubles his speed, how will the force acting on his bicycle towards the center will change?

Solution:

Centripetal force is given as,

[Tex]\sum F= Mass. Acceleration = ma_r= \frac{mv^2}{r} [/Tex]

It is observed that,

F∝ v²

Therefore, if the velocity is doubled, the force acting on the bicycle will become 4 times than before.

Question 3: A plane is flying with a speed of 120 m/sec, it makes a turn to join a circular path leveling with the ground. What will be the radius of the circular path formed if the centripetal acceleration is equal to the acceleration due to gravity?

Solution:

Centripetal acceleration (a_c ) is equal to the acceleration due to gravity (g), which is assumed to be 10m/s²

speed of plane(v) = 120m/s

Using the formula for centripetal acceleration:

a_c =v²/r

given that a_c = g = 10m/s²

r = v²/a_c

Let’s assume the acceleration due to gravity to be 10 m/sec²

r = (120)² / 10

⇒ r = 14400 / 10

⇒ r = 1440m

Question 4: What is the device used for measuring the speed of rotation in electrical machines or in objects moving in uniform circular motors?

Solution:

Tachometer is the device which is used to measure the speed of rotation in electrical machines or in objects moving in uniform circular motion.

Question 5: The property of conservation of energy is applied when an object moves in a uniform circular motion. How?

Solution:

Conservation of Energy is a universal fact and this is properly applied when an object moves in uniform circular motion, when the object moves in uniform circular motion, the speed remains constant and hence, the kinetic energy remains constant. however, due to the constant change in velocity, the momentum keeps on changing.

Question 6: Suppose you are sitting in a room and felt the need to increase the speed of the fan, you increased the regulator and the blades accelerated at 2 rad/sec². It accelerated for 3 seconds and the final angular frequency of the blades became 7 rad/sec. What was the initial angular frequency?

Solution:

Formula for Angular velocity is, 

α = Δw/ Δt

Given: 

• Time for which the blades accelerated = 3 seconds.
• Acceleration = 2 rad/sec²

⇒ 2 ={w_f – w_i}/ Δt

⇒ 2 = {7 – w_i} / 3

⇒ 7- w_i = 6

⇒ w_i = 1 rad/sec

FAQs on Uniform Circular Motion

What is Difference between Uniform Circular Motion and Non-Uniform Circular Motion?

The only key difference between uniform and non-uniform circular motion is the speed of the motion, i.e., in uniform circular motion speed of the object performing is constant whereas, in a non-uniform circular motion, the speed is varying.

Why Uniform Circular Motion is called Accelerated Motion?

In uniform circular motion speed is constant but due to the changing direction of the motion velocity is always changing thus there is always a change in velocity i.e., there is always an acceleration acting on the object.

What is the Centripetal Force?

Due to the always present acceleration, there is a force acting on the body in the direction of the center of the circular path which is called centripetal force.

What are Examples of Uniform Circular Motion?

The motion of electrons in Bohr’s atomic model, constant revolutions of a ceiling fan, constant spinning of CD and DVD in the player, and motion of a Ferris wheel are some examples of uniform circular motion.

Can an Object be in a Uniform Circular Motion if its Speed is Changing?

No, as uniform circular motion is defined for the constant speed of the object if speed started changing it becomes non-uniform circular motion.