# 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 m2

Wanted: The rotational kinetic energy (KE)

Solution :

The formula of the rotational kinetic energy :

KE = 1/2 I ω2

KE = the rotational kinetic energy (kg m2/s2), I = the moment of inertia (kg m2), ω = the angular velocity (rad/s)

The rotational kinetic energy :

KE = 1/2 I ω2 = 1/2 (1)(2)2 = 1/2 (1)(4) = 2 Joule

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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 r2

I = the moment of inertia (kg m2), 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 m2

The rotational kinetic energy of the pulley :

KE = 1/2 I ω2 = 1/2 (0.4)(4)2 = (0.2)(16) = 3.2 Joule

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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 r2

I = moment of inertia (kg m2), 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 m2

The rotational kinetic energy of the ball :

KE = 1/2 I ω2 = 1/2 (0.04)(10)2 = (0.02)(100) = 2 Joule

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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 r2 = (0.5)(0.1)2 = (0.5)(0.01) = 0.005 kg m2

The rotational kinetic energy :

KE = 1/2 I ω2 = 1/2 (0.005)(2)2 = 1/2 (0.005)(4) = (0.005)(2) = 0.01 Joule