# Circular motion – problems and solutions

1. A 10-kg object moves in a circle at constant speed 4 m/s. If the radius of the circle is 0.5 meters, then :

1) The frequency of circle is 4/π Hz

2) The centripetal acceleration is 32 m.s^{-2}

3) The centripetal force is 320 N

4) The period is 4π s.

Which are the true statements?

__Known :__

Mass of object (m) = 10 kg

The linear velocity (v) = 4 m/s

The radius of circle (r) = 0.5 meters

__Solution :__

1) The frequency of circle

v = 2 π r f

4 = 2 π (0.5) f

4 = π f

f = 4/π Hertz

2) The centripetal acceleration

a_{s} = v^{2} / r = 4^{2} / 0,5 = 16 / 0,5 = 32 m/s^{2 }

3) The centripetal force

F = m a_{s} = (10)(32) = 320 N

4) Period

T = 1 : f = 1 : 4/π = 1 x π/4 = π/4

2. An object moving in a circle of radius 6 meters. If the object rounds 16 circles in 2 minutes, what is the linear velocity of the object.

__Known :__

Radius (r) = 6 meters

The angular velocity (ω) = 16 revolutions / 2 minutes = 8 revolutions / minutes = 8 revolutions / 60 seconds = 0.13 revolutions/second.

__Wanted:__ The linear velocity (v)?

__Solution :__

v = r ω = (6 meters)(0.13 revolutions/second) = 0.8 meters/second

In radian :

1 revolution = 2π radian = 2(3.14) = 6.28 radian

The angular velocity = 8 (6.28) radians / 60 seconds = 50.24 radians / 60 seconds = 0.84 radians/second

v = r ω = (6 meters)(0.84 radians/second) = 5.04 radians/second.

3. An object with radius of 20/π cm rotates 4 times in 1 second. What is the linear velocity of the edge of object.

__Known :__

Radius (r) = 20/π cm = 20 / 3.14 cm = 6.4 cm = 0.064 meters

The angular velocity (ω) = 4 revolutions / 1 second = 4 revolutions / second.

1 revolution = (2)(3.14) radians = 6.28 radians

The angular velocity (ω) = (4)(6.28) radians/second = 25.12 radians/second

__Wanted :__ The linear velocity of the edge of object (v)

__Solution :__

v = r ω = (0.064 meters)(25.12 radians/second) = 1.6 meters/second

4. An object moving in a circle at constant speed, the linear velocity of the object depends on…

Solution :

The equation of the linear velocity of the circular motion :

v = the linear velocity

d = 2πr = circumference

T = period = the time required for one complete revolution.

5. An object moving in a circle of radius 50 meters. If the angular speed of the object is 120 rpm, what are the time interval and the linear velocity of the object?

__Known :__

Radius (r) = 50 cm = 0.5 meters

The angular velocity (ω) = 120 rpm = 120 revolutions / 1 minute = 120 revolutions / 60 minutes = 2 revolutions / 1 second

1 revolution = 2π radian

The angular velocity (ω) = 2 (2π radians) / 1 second = 4π radians/second

__Wanted :__ The time interval (T) and the linear speed (v)

__Solution :__

Period (T) :

Period is the time required for one complete revolution.

An object rotates two revolutions in 1 second = 1 revolutions per 0.5 seconds. Period = 0.5 second.

The linear velocity (v) :

v = r ω = (0.5 meters)(4π radians/second) = 2π meters/second.