# A flywheel is a solid disk that rotates about an axis that is perpendicular to the disk at its center. Rotating flywheels provide a means for storing energy in the form of rotational kinetic energy and are being considered as a possible alternative to batteries in electric cars. The gasoline burned in a 300-mile trip in a typical midsize car produces about 1.2*10^9 J of energy. How fast would a 13-kg flywheel with a radius of 0.30m have to rotate to store this much energy? Give your answer in rev/min.

## The kinetic energy stored in a rotating object is

(1/2) I w^2

I is the moment of inertia of the disk about an axis perpendicular to the disk at the center, which you can look up (If they want you to derive it reply to me that you need a derivation)
I = (1/2) M R^2 where R is the radius of the flywheel and M is the mass
w is the angular velocity in radians per second
w = 2 pi (revolutions per second)
so set your Joules equal to (1/2) I w^2 and solve for w
then change w from radians per second to revs per minute
revs/min = w in rad/sec * 1 rev/2pi radians * 60 seconds/1 minute

## To solve this problem, you need to determine the speed at which the flywheel must rotate to store the given amount of energy.

The formula for the rotational kinetic energy of a flywheel is given by:

E = (1/2) I ω^2

Where:
E is the rotational kinetic energy
I is the moment of inertia of the flywheel
ω is the angular velocity of the flywheel

The moment of inertia of a solid disk is given by:

I = (1/2) m r^2

Where:
m is the mass of the flywheel
r is the radius of the flywheel

You are given the energy (E) and the mass (m) of the flywheel. To find the angular velocity (ω), we need to rearrange the equation for angular velocity:

ω = √(2E / I)

Let's plug in the values given in the problem:

E = 1.2 * 10^9 J
m = 13 kg
r = 0.30 m

To find the angular velocity in rev/min, we need to convert it from radians per second to revolutions per minute. There are 2π radians in one revolution and 60 seconds in one minute.

Do you want me to calculate the angular velocity for you?