# Rotational kinetic energy – problems and solutions

1. An object has the moment of inertia of 1 kg m^{2 }rotates at a constant angular speed of 2 rad/s. What is the rotational kinetic energy of the object?

__Known :__

The moment of inertia (I) = 1 kg m^{2}

The angular velocity (ω) = 2 rad/s

__Wanted:__ The rotational kinetic energy (KE)

__Solution :__

The formula of the rotational kinetic energy :

KE = 1/2 I ω^{2}

*K**E **= **the rotational kinetic energy **(kg m*^{2}*/s*^{2}*), I = **the moment of inertia **(kg m*^{2}*), **ω **= **the angular velocity **(rad/s)*

The rotational kinetic energy :

KE = 1/2 I ω^{2} = 1/2 (1)(2)^{2 }= 1/2 (1)(4) = 2 Joule

2. A 20-kg cylinder pulley with a radius of 0.2 m rotates at a constant angular speed of 4 rad/s. What is the rotational kinetic energy of the pulley?

__Known :__

Mass of cylinder pulley (m) = 20 kg

The radius of cylinder (r) = 0.2 m

The angular speed (ω) = 4 rad/s

__Wanted :__ What is the rotational kinetic energy

__Solution ;__

Formula of the moment inertia of cylinder :

I = 1/2 m r^{2 }

*I = **the moment of inertia (**kg m*^{2}*), m = mass (kg), **r = **radius** (meter)*

The moment of inertia of cylinder pulley :

I = 1/2 (20)(0.2)^{2} = (10)(0.04) = 0.4 kg m^{2}

The rotational kinetic energy of the pulley :

KE = 1/2 I ω^{2} = 1/2 (0.4)(4)^{2 }= (0.2)(16) = 3.2 Joule

3. A-10 kg ball with radius of 0.1 m rotates at a constant of 10 rad/s. What is the kinetic energy of the ball.

__Known :
__

Mass of ball (m) = 10 kg

Radius of ball (r) = 0.1 m

Angular velocity (ω) = 10 rad/s

__Wanted :__ The rotational kinetic energy

__Solution :__

Formula of the moment of inertia :

I = (2/5) m r^{2 }

*I = **moment of inertia** (kg m*^{2}*), m = mass (kg), **r = **radius **(m)*

Moment of inertia of the ball :

I = (2/5)(10)(0.1)^{2} = (4)(0.01) = 0.04 kg m^{2}

The rotational kinetic energy of the ball :

KE = 1/2 I ω^{2} = 1/2 (0.04)(10)^{2 }= (0.02)(100) = 2 Joule

4. A 0.5-kg particle rotates at a constant angular speed of 2 rad/s. What is the rotational kinetic energy of the particle if the radius of circle is 10 cm.

__Known :__

Mass of particle (m) = 0.5 kg

The radius of ball (r) = 10 cm = 10/100 = 0.1 m

The angular speed (ω) = 2 rad/s

__Wanted :__ The rotational kinetic energy

__Solution :__

Moment of inertia for particle :

I = m r^{2 }= (0.5)(0.1)^{2} = (0.5)(0.01) = 0.005 kg m^{2}

The rotational kinetic energy :

KE = 1/2 I ω^{2} = 1/2 (0.005)(2)^{2 }= 1/2 (0.005)(4) = (0.005)(2) = 0.01 Joule