# Uniform circular motion – problems and solutions

1. An object moves in a circle with the constant angular speed of 10 rad/s. Determine (a) Angular speed after 10 seconds (b) Angular displacement after 10 seconds.

__Known :__

Angular speed (ω) =10 rad/s

__Wanted :__

(a) Angular speed (ω) after 10 seconds.

(b) Angle (θ) after 10 seconds

__Solution :__

(a) Angular speed (ω) after 10 seconds

Object in uniform circular motion so that angular speed is constant, 10 rad/s.

(b) Angular displacement (θ)

Constant angular speed 10 radians/second means the object around 10 radians each second. After 10 seconds, the object around 10 x 10 radians = 100 radians.

2. A particle moves in a circle with the constant speed of 10 m/s. Radius of circle = 1 meter. Determine (a) Particle’s speed after 5 seconds (b) Particle’s displacement after 5 seconds (c) Centripetal acceleration.

__Known :__

Radius of circle (r) = 1 meter

Particle’s speed (v) = 10 m/s

__Solution :__

(a) __ Particle’s speed after 5 seconds__

The motion of object is in the uniform circular motion so that speed is constant, 10 m/s.

(b)__ Particle’s displacement after 5 seconds__

10 meters/second means each 1 second, particle’s displacement = 10 meters. After 5 seconds, particle’s displacement = 5 x 10 meters = 50 meters.

(c)__ Centripetal acceleration (a___{r}__)__

a_{r} = v^{2} / r = 10^{2} / 1 = 100 / 1 = 100 m/s^{2}

3. A ball attached to one end of a cord, is revolved in a circle with radius of 2 meters at the constant speed of 60 rpm. Determine (a) the magnitude of the angular speed after 2 seconds (b) the angular displacement after 1 minute.

__Known :__

Radius of circle (r) = 2 meters

Angular speed (ω) = 60 rpm = 60 revolutions / 1 minute

= 60 revolutions / 60 seconds = 1 revolution / second = 2π radians / second

= 2(3.14) radians / second= 6.28 radians / second

__Solution :__

(a) __Angular speed (ω) after 2 seconds__

The angular speed is constant so after 2 seconds, the angular speed (ω) = 6.28 radians / second

(b) __Angular displacement (θ) __

The angular speed = 1 revolution/second means each 1 second, ball experience 1 revolution. After 60 seconds, ball moves 60 revolutions.

The angular speed = 6.28 radians/second means each 1 second, the ball moves with the angle of 6.28 radians. After 60 seconds, the ball moves 376.8 radians.

4. A bike wheel rotates 120 revolutions in 60 seconds. What is the angular speed?

__Solution :__

__ (a) revolutions per minute (rpm)__

120 revolutions / 60 seconds = 120 revolutions / 1 minute = 120 revolutions / minute = 120 rpm

(b) __degrees per second (__^{o}__/s)__

1 revolution = 360^{o}, 120 revolutions = 43200^{o}

120 revolutions / 60 seconds = (120)(360^{o}) / 60 seconds = 43200^{o} / 60 seconds = 720^{o}/second

(c) __radians per second (rad/s)__

1 revolution = 6.28 radians

120 revolutions / 60 seconds = (120)(6.28) radians / 60 seconds = 753.6 radians / 60 seconds = 12.56 radians/second.

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