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Storing energy in flywheels. 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 fly wheels. Suppose we want to build such a flywheel in theshape of a hollow cylinder of inner radius 0.500 $\mathrm{m}$ and outer radius $1.50 \mathrm{m},$ using concrete of density $2.20 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .$ (a) 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 2.5 $\mathrm{MJ}$ of energy in its rotational motion? (b) Suppose that by strengthening the frame you could safely double the flywheel's rate of spin. Whatlength of flywheel would you need in that case? (Solve this part without reworking the entire problem!)

a) 22.5 $\mathrm{m}$b) 5.63 $\mathrm{m}$

Physics 101 Mechanics

Chapter 9

Rotational Motion

Physics Basics

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

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In the first part of this question, we're going to help lead the lint off off Live, which is represented by L. In order to kill. Let the let off off la Ville We have to right love William off that flabby, which is week Will stow pile into are two square minus R one square. No, let's write the mess off the flyby in Trump's off density And William my articles, too. Dance team in to blow away, will you? While sitting the radio off William, we into this equation, we will get M equals two row by L into our two square minus r one square. Here are two is the altar radius off the wheel and are one Is that in the radius off the wheel. So call it equation one no lifts, right. The moment off inertia for this wheel equals two one by two and into our one squared Plus are too square now, City we were you off this having to this question? We will get equals to one by two row pile into our two square minus r one square into are one squared plus or two square. So this will give us Michael's to one by two Rope I l in tow are tourists are four minus are wondrous part for so call it equated to now let's define the rotational kinetic energy for this flywheel care rotational equals to one by two Iomega Square Now sitting the world off I into this equation we will get traditional equals two one by two months Flower one by two Rafael into our tourist part four minus our oneness part for into my God square. No soloing this question for l. We can right l equals two for conditional divided by rope I into our tourists. Part four minus one is powerful into my that square more setting the world is into this equation. We will get l equals two foreign to point six month black way Turner's for six. Joel, you wanted way 2.20 multiplied by tennis Part six Katie. Thank you into 3.14 into 1.5 Medio Holder is beautiful minus 0.5 meter holders bar for into to pirate And he wanted by 60 seconds old square. So this will give us l equals two 22.5 meter. Let's come toward the second part of this question in this part of the question, we are again going to kill Let the land when we doubled our angler velocity. So from this equation, we can see that this land is enormously proportional to the square off this angler velocity. So when we will double this England velocity, the length will decrease by four times so we can write. So w prime pickles too are going toe w then prime will be one before it so we can write. So the Cleveland tell prime will be cause to 22.5 meter divided by four, which is a good school. So prime will be calls, too. Five point sit three meter. So end off this question.

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