Question

You are designing a flywheel to store kinetic energy. if all of the following uniform objects have the same mass and same angular veloc- ity, which one will store the greatest amount of kinetic energy? which will store the least? explain.

(a) a solid sphere of diameter d rotating about a diameter;
(b) a solid cylinder of diameter d rotating about an axis perpendicular to each face through its center;
(c) a thin-walled hol- low cylinder of diameter d rotating about an axis perpendicular to the plane of the circular face at its center;
(d) a solid, thin bar of length d rotating about an axis perpendicular to it at its center.

Answer 1

The thin-walled hollow cylinder will store the greatest amount of kinetic energy because it has the highest moment of inertia, while the solid, thin bar will store the least due to its smaller moment of inertia.

Step-by-step explanation:

When considering which object will store the greatest amount of kinetic energy, we look at the rotational kinetic energy formula KE = ½ Iω2 where I is the moment of inertia and ω is the angular velocity. Since all given objects have the same mass and angular velocity, the key factor is the moment of inertia, which depends on the mass distribution relative to the axis of rotation. The more mass that is distributed away from the axis, the higher the moment of inertia and therefore the more kinetic energy the object can store.

(a) A solid sphere rotating about a diameter will have the moment of inertia I = ⅖Md2/8, where M is the mass and d is the diameter.
(b) A solid cylinder rotating about its central axis has I = ½MR2, where R is the radius.
(c) A thin-walled hollow cylinder will have I = MR2.
(d) A solid, thin bar with length d and rotating about its center has I = ⅖Md2/12.

Comparing the moments of inertia, we can see that the thin-walled hollow cylinder (c) will store the greatest amount of kinetic energy because it has the largest moment of inertia for a given mass and radius. The solid, thin bar (d), in contrast, will store the least amount of kinetic energy due to having the smallest moment of inertia among the options provided.