# Numerical problem on Rotational Kinetic Energy

Last updated on July 12th, 2023 at 03:56 pm

In this post, we will review the formula of Rotational Kinetic Energy and then solve **Numerical problems based on Rotational Kinetic Energy.**

## Rotational Kinetic Energy formula from the linear kinetic energy formula

The equation for linear kinetic energy =

KE_{L}= (1/2) m v^{2}Let’s Convert that equation to its angular analog:

KE = (1/2) (mr^{2}).(v^{2}/r^{2}) = (1/2) (mr^{2}).(v/r)^{2}= (1/2) I ω^{2}KE_{R}= (1/2) I ω^{2}

The equation of linear kinetic energy can be converted to the angular analog equation when the motion is rotational. The angular velocity ω takes the place of the linear velocity v, and the momentum of inertia (**I)** takes the place of the mass m.

You may wish to read a separate post specifically on the Detailed step-by-step derivation of the Rotational KE formula for class 11.

**Numerical problem based on Rotational Kinetic Energy**

While solving the Numericals you can have a quick look at this post:

Moment of Inertia of different bodies

**1 ) A 100-kg solid sphere with a radius equal to 2.0 m is rotating at 10.0 radians/s, what is its rotational kinetic energy?**

**Solution:**

ω = 10 rad/sec

r= 2 m

m = 100 kg

For a solid sphere, the momentum of inertia I = (2/5) m r

^{2}

KE= (1/2) (2/5) m r_{R}= (1/2) I ω^{2}^{2}.ω= (1/2) (2/5) 100 2^{2}^{2}.10= 800 J^{2}

**2 )** **How much rotational kinetic energy does a spinning tire of mass 10.0 kg and radius 0.50 m have if it’s spinning at 40.0 rotations/s?**

**Solution:**

ω = 40 rps = 40 x 2π rad/sec =

80πrad/secr= 0.5 m

m = 10 kg

For a

spinning tire, the momentum of inertia I = m r^{2}

KE= (1/2) m r_{R}= (1/2) I ω^{2}^{2}.ω= (1/2) 10 (0.5)^{2}^{2}.()80π= 78.8 x 10^{2}^{3}J

To revise the formulas you can have a quick look at this post: **Moment of Inertia of different bodies**

**3 ) How much rotational kinetic energy does a spinning tire of mass 12 kg and radius 0.80 m have if it’s spinning at 200.0 radians/s?**

**4 )How much work do you do to spin a tire, which has a mass of 5.0 kg and a radius of 0.40 m, from 0.0 radians/s to 100.0 radians/s?**

**5 )How much work do you do to spin a hollow sphere, which has a mass of 10.0 kg and a radius of 0.50 m, from 0.0 radians/s to 200.0 radians/s?**