## 15 Common Problems of Circular Motion

- Category: High School
- Written by fisikastudycenter

physics.fisikastudycenter.com - Learning circular motion in 15 common problems. Expressing quantities of circular motion in different units, angular speed, angular position, centripetal acceleration, centripetal force and some gears relationships will be discussed.

**Problem 1**

Express in radians :

a) 90^{o}

b) 270^{o}

**Solution**

360^{o} = 2π radians

a) 90^{o}

b) 270^{o}

**Problem 2**

Express in rad/s :

a) 120 rpm

b) 60 rpm

**Answer**

1 rpm = 1 revolution per minute

1 revolution equals 2π radians or

1 revolution equals 360^{o}

1 minute equals 60 sec

a) 120 rpm

b) 60 rpm

**Problem 3**

A particle moves at 50π rad/s of angular speed. Find the frequency of the particle motion!

**Answer**

**Problem 4**

The angular speed of a particle in circular motion is 12 rad/s. Find the speed of particle if the radius = 2 m!

**Answer**

**Problem**** 5**

A mass of 1 kg rotates with an angular speed of 120 rpm. If its radius is 2 meter then find the centripetal acceleration of this particle!

**Answer**

Data :

ω = 120 rpm = 4π rad/s

r = 2 meter

m = 1 kg

a_{sp} = ...?

a_{sp} = ^{V2}/_{r} = ω^{2} r

a_{sp} = (4π)^{2} (2) = 32π^{2} m/s^{2}

**Problem 6**

Find the centripetal force acting on a body of 1 kg in a circular motion with its radius of 2 m and speed of 3 m/s!

**Answer**

Data :

m = 1 kg

r = 2 meter

V = 3 m/s

F_{sp} = ....?

F_{sp} = m ( ^{V2}/_{r} )

F_{sp} = (1)( ^{32}/_{2} ) = 4,5 N

**Problem**** 7**

Given two sprockets of a bicycle as shown below!

The first sprocket radius and the second are 20 cm and 10 cm respectively and the angular speed of the first sprocket is 50 rad/s, find the second sprocket angular speed!

**Answer**

Data :

r_{1} = 20 cm

r_{2} = 10 cm

ω_{1} = 50 rad/s

ω_{2} = ...?

The two sprockets have the same in linear speeds if the chain does not slip or stretch so then :

**Problem**** 8**

Two rotating gears!

The speed of the first gear is 20 m/s with radius of the first and the second gears are 20 cm and 10 cm, find the second gear speed!

**Answer**

The two gears in the obove relationship have the same in angular velocities:

**Problem**** 9**

Three gears are configurated as below!

The radius are :

r_{1} = 20 cm

r_{2} = 10 cm

r_{3} = 5 cm

If the angular speed of the firs gear is 100 rad/s, find the angular speed of the third gear!

**Answer**

**Problem 10 **

A particle in a circular motion has its angular speed of 4 rad/s. Determine the angular position at t = 5 sec!

**Answer**

Finding the angular position:

θ = ωt

θ = (4)(5) = 20 radian.

**Problem 11 **

A particle in circular motion initially at rest. The particle angular acceleration is 2 rad/s^{2}. Find :

a) the angular speed at t = 5 sec

b) the angular position at t = 5 sec

**Answer**

Data :

α = 2 rad/s^{2}

ω_{o} = 0

t = 5 sekon

Uniformly accelerated circular motion

a) ω_{t} = ω_{o} + αt

ω_{t} = (0) + (2)(5) = 10 rad/s

b) θ = ω_{o}t + ^{1}/_{2} αt^{2}

θ = (0)(5) + ^{1}/_{2} (2)(5)^{2}

**Problem 12**

A car of 2000 kg of mass at 20 m/s of speed travels in circular motion with its radius of 100 m.

Find the normal force acting on the car in the peak point (the highest)!

**Answer**

Free body diagram of forces at the highest point :

Newton's law :

**Problem 13**

A particle in circular motion with its radius of 50 cm shown below.

If the mass of particle is 200 gram and the acceleration due to gravity is 10 m/s^{2}, find the tension of the rope when it is in the highest point!

**Answer**

Newton's law for rotating bodies:

The free body diagram shown

So then :

**Problem 14**

Use the previous problem, find the tension of the rope when it is in the lowest point!

**Answer**:

**Problem 15**

Find the angular speed of the second gear!

**Answer**