Uniform Circular Motion

The term “Uniform circular motion” is the kind of motion of an object in a circle at a constant speed. With uniform circular motion, as an object moves in a particular circle, the direction of it is constantly changing. Herein, in all instances, the object is moving tangent to the circle. Let us study this angular motion in detail.

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Circular Motion

Circular motion is the movement of an object in a circular path.

Uniform Circular Motion

This motion refers to the circular motion if the magnitude of the velocity of the particle in circular motion remains constant. The non-uniform circular motion refers to the circular motion when the magnitude of the velocity of the object is not constant. Another special kind of circular motion is when an object rotates around itself also known by spinning motion.

Variables in Circular Motion

Angular Displacement 

The angle which is subtended by the position vector at the center of the circular path refers to the angular displacement.

Angular Displacement (Δθ) = (ΔS/r)

Where Δ’s refers to the linear displacement while r is the radius. Radian is the unit of Angular Displacement.

Browse more Topics Under Motion In A Plane

• Introduction to Motion in a Plane
• Scalars and Vectors
• Resolution of Vectors and Vector Addition
• Addition and Subtraction of Vectors – Graphical Method
• Relative Velocity in Two Dimensions
• Uniform Circular Motion
• Projectile Motion

Angular Acceleration 

It refers to the rate of time of change of angular velocity (dῶ).

Angular acceleration (α) = dῶ/dt = d2θ / dt2

Its unit is  r_ad/s₂  and dimensional formula [T]^-2. The relation between linear acceleration (a) and angular acceleration (α)

A = rα, where r is the radius.

Angular Velocity

It refers to the time rate change of angular displacement (dῶ).

Angular Velocity (ῶ) = Δθ/Δt

Angular Velocity is a vector quantity. Its unit is rad/s. The relation between the linear velocity (v) and angular velocity (ῶ) is

v = rῶ

Centripetal Acceleration

It refers to an acceleration that acts on the body in circular motion whose direction is always towards the center of the path.

Centripetal Acceleration (α) = v₂/r = rῶ2.

The magnitude of this acceleration by comparing ratios of velocity and position around the circle. Since the particle is traveling in a circular path, the ratio of the change in velocity to velocity will be the same as the ratio of the change in position to position. It is also known as radial acceleration as it acts along the radius of the circle. Centripetal Acceleration is a vector quantity and the unit is in m/s2.

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Solved Question for You

Q. During the course of a turn, an automobile doubles its speed. How much must additional frictional force the tires provide if the car safely makes around the curve? Since Fc varies with v2, an increase in velocity by a factor of two must be accompanied by an increase in centripetal force by a factor of four.

A satellite is said to be in geosynchronous orbit if it rotates around the earth once every day. For the earth, all satellites in geosynchronous orbit must rotate at a distance of 4.23×107 meters from the earth’s center. What is the magnitude of the acceleration felt by a geosynchronous satellite?

Solution:  The acceleration felt by any object in uniform circular motion is given by

a = \( {v²}\div{r} \)

We are given the radius but must find the velocity of the satellite. We know that in one day, or 86400 seconds, the satellite travels around the earth once. Thus:

v = \( {Δr}\div{Δt} \) =  \( {2πr}\div{Δt} \)   = \( {2π × 4.23 × 10^7}\div{86400} \) = 3076m/s

a = \( {v^2}\div{r} \) = \( {3076^2}\div{(4.23 × 10^7)} \)  = .224m/s²