Uniform circular motion: Easy concise discussion with 5 examples

Table of Contents

Uniform Circular Motion refers to the motion of an object traveling at a constant speed along a circular path. The object’s speed remains constant. However, its velocity is constantly changing. This change occurs due to the continuous change in direction as the object moves along the curve.

Key characteristics of uniform circular motion includes:

Constant Speed: The object moves with a constant speed along the circle. However, because the direction is always changing, the velocity, which is a vector, changes continuously.
Centripetal Force: An object follows a circular path only if a force acts toward the circle’s center. This is called the centripetal force.

Consider a particle moving along a circular path when it moves from point A to point B as in figure below.

The reference point OA makes an angle with the line OB that represents the line joining the new position of the object and the center of the circle. The object has covered a distance S along the circle making an arc with θ.

Angular displacement is the angle swept by a line joining end of an object in a circular path with the center of the path when it moves from one point to another in a circular motion.

Angles in circular motion are usually expressed in radians θ^c.

$$\text{Angle θ in radians} = \frac{arc length S (AB)}{radius r (OA)}$$

That is:

$$θ (radians)= \frac{S}{r}$$

it therefore follows that S = rꝊ^c

A radian is defined as an angle subtended at the center of a circle by an arc length equal to the radius of the circle.

Therefore angle θ subtended by the circumference at the center of a circle of radius r is therefore given by;

$$θ = \frac{circumference} {radius}$$

and we can write circumference in terms of π such that:

$$\frac{2\pi r}{r} = 2\pi^c$$

We can relate the degrees from 2π^c = 360^o .

Problem

An object traces an arc of length 10.98 while attached to a cord of length 3.2 m that is fixed on a fixed surface on a flat smooth surface. Determine the angular displacement by the object.

Solution

We visualize the setup as in figure below:

from the equation S = rꝊ^c

$$θ^c = \frac{S}{r} = \frac{10.98}{3.2} = 3.43 \ radians $$

if we can express answers in degrees, then 3.43 radians = 196.52^o .

Practice Question

An object moves a distance of 80.12 π along a circular path of radius 3.8m. Determine it’s angular displacement.

Angular velocity for a uniform circular motion

Angular velocity is defined as the rate of change of angular displacement with time.

we can shorten the equation by using symbols alone:

$$\omega = \frac{\Delta \theta}{\Delta t} $$

The SI Units of angular velocity is radians per second (rads-1)

Consider the equation that relates angular displacement in radians with the arc length made by the object:

$$\theta = \frac{S}{r} $$

to get the rate of change of speed, we divide both sides with t as they both represent displacement of the object.

$$\frac{ \theta} {t} = \frac{S}{rt} = \frac{1}{r}(\frac{S}{t})$$

but distance s when divided by time gives velocity. That is;

$$v= \frac{ S}{t}$$

where v is the linear velocity representing the velocity of the object along the circular path.

$$\omega = \frac{\theta}{t}$$

Hence the equation above becomes:

hence, the angular velocity can be expressed in terms of linear velocity and radius as show in equation below:

$$\omega = \frac{v}{r}$$

similarly, linear velocity can be expressed in terms of angular velocity as v=ωr.

An object in circular motion has both linear and angular velocity. The time taken to make one complete circle is called the periodic (T) and is given by T= circumstances/speed.

Let us now consider the time taken for a body to make one complete circle in a circular path. At that one circle, it will have covered the circumference of a circle. The time taken to complete such one revolution is called periodic time (T).

from the equation :

$$ time = \frac{distance}{speed}$$

and that:

$$T = \frac{circumference}{speed}$$

and circumference = 2πr, hence

$$T =\frac{2\pi r}{\omega r} = \frac{2\pi}{\omega}$$

Hence T in the above equation becomes:

$$T = \frac{2 \pi}{\omega}$$

but

$$\frac{1}{T} = \frac{\omega}{2 \pi}$$

therefore:

$$\omega = \frac{2 \pi}{T}$$

but since:

$$\frac{1}{f} = T$$

$$\omega = \frac{2 \pi}{\frac{1}{f}} = 2 \pi f$$

Example question

A metallic ball is whiled in a horizontal circle making 5 revolution s per second.Determine:

Period T
angular velocity and
linear speed v

Centripetal force

Consider on object m held by a cord om positioned at A. The Object is whirled in a circular motion and after some time Δt, the object is at position B. The velocity of the object in linear direction changes from V_A to V_B. If there was no force acting on the body, the object will not change directions but will go in a straight line. There must be a force that maintain the body at a constant distance distance from the center o.

Centripetal force F_c refers to the force that keeps a body in circular motion. A body in a circular motion is accelerating and from newton’s second law of motion, there must be a force acting on it to cause acceleration. Centripetal force is usually directed towards the center of the circular path. The Centripetal force is the force responsible for the constant change of direction otherwise the body would naturally follow a straight line if there was no force acting to keep the body in circular motion.

The value of the centripetal force is derived from newton’s second law of motion which states that: the rate of change of momentum of a body is directly proportional to the resultant force in the direction of force.

Momentum means mass multiplied by velocity.

Because velocity of a body in circular motion is changing, it’s momentum must also be changing.

The newton’s second law can be described as F=ma, where a = acceleration and m is the mass.

but the acceleration of the body of the body is given by a= v²/r, where v is the linear speed of the object while it is in circular motion. hence

From definition of angular velocity we had shown that, ω is given by v/r, and hence v=ωr.

it follows that F_c = m(ωr)²/r = mω²r²/r = mω²r .

Tension

If a body is attached to a string and swung around on a horizontal circle, the centripetal force that keeps the body in the circular orbit is kept as tension in the string. For the body to remain in circular motion, the centripetal force is equal to the tensional force.

From the equation F_c = mω²r , it shows that centripetal force is directly proportional to the angular velocity meaning that a larger force will be required to maintain the body in motion if it is swung faster.