It has been suggested that we should use our power plants to generate energy in

Question

It has been suggested that we should use our power plants to generate energy in the off-hours (such as late at night) and store it for use during the day. One idea put forward is to store the energy in large flywheels. Suppose we want to build such a flywheel in the shape of a hollow cylinder of inner radius 0.490m and outer radius 1.60m , using concrete of density 2150 km/m^3. If, for stability, such a heavy flywheel is limited to 1.75 second for each revolution and has negligible friction at its axle, what must be its length to store 2.50MJ of energy in its rotational motion? Suppose that by strengthening the frame you could safely double the flywheel's rate of spin. What length of flywheel would you need in that case? (Solve this part without reworking the entire problem!)

Explanation / Answer

IN MY ANSWER ABOVE I FORGOT TO INCLUDE THAT PI IN THE CALCULATION!! THE ANSWERS HERE SHOULD BE OK...

momen of inertia of a cylinder: I = 1/2 m r^2 = 1/2 * (density * length * pi r^2) r^2

= 1/2 density * L * r^4

for your hollow cylinder...

I = 1/2 * 2150 * pi * L * (1.60^4 - 0.490^4) = 21938 L

Now... energy = (1/2) I w^2 where

w = 2pi radians / 1.75 sec = 3.59 rad/sec

so

2500000 = 1/2 * 21938 L * 3.59^2

and the length of the cylinder is   L = 17.68 meters

(check my calculations if, for some reason, this doesnt come out right... but I think it's straight)

if the rate of spin is doubled, the length would be divided by 4 since the length is inversely proportional to w squared. So

new length = 17.68 / 4 = 4.42 m

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