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In redesigning a piece of equipment, you need to replace a solid spherical part of mass $M$ with a hollow spherical shell of the same size. If both parts must spin at the same rate about an axis through their center, and the new part must have the same kinetic energy as the old one, what must be the mass of the new part in terms of $M ?$

$\frac{3}{5} \mathrm{M}$

Physics 101 Mechanics

Chapter 9

Rotational Motion

Physics Basics

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

University of Michigan - Ann Arbor

Simon Fraser University

Hope College

University of Winnipeg

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me. Let's write down the two moments of inertia of the two objects. So I, friend Sphere, which I'm going to know with no sub scripts, is equal to 2/5 times master the sphere again off no sub scripts times are squared, whereas I for the shell, which are the known this way, is equal to 2/3 times mass of the show. Times are squared these two things I got from Table 92 and they're one of the common moments of inertia of objects. So they're tabulated. So since the kinetic energy of a rotating object is equal to 1/2 times I time saying, Oh, velocity squared the same kinetic energy and the same in your velocity for two objects implies that their moment of inertia is must be the same. And so I was evil. So I shall. We're told they have the same connect energy and the same angular velocity and the problem so we can deduce us from that. Now we just need to plug in the two moments of the nursery that we have over here. So let's get his two fists. Omar squared is equal to 2/3 him. Shell R squared. Remember, they have the same radius, so these are exactly the same. And so they cancel out the twos, cancel out, and we're solving for M Shell in terms of elm. So I'm going to multiply both sides by three. So I get him. Shell is equal to 3/5 time Sam, and that's final answer here.

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