Practice Problem 9.10 In a common lecture demonstration, an instructor races var

Question

Practice Problem 9.10 In a common lecture demonstration, an instructor races various round objects by releasing them from rest at the top of an inclined plane and letting them roll down the plane (Figure 1). Before the objects are released, the students guess which object will win. How much faster is the solid cylinder moving at the bottom of the ramp than the hollow cylinder? Part A - Practice Problem: Find the ratio of the final speeds for a hollow sphere and a solid sphere. Express your answer to four significant figures. Part B - Practice Problem: What is the ratio of the rotational kinetic energy of the hollow sphere to the solid sphere at the bottom of the ramp? Express your answer to three significant figures.

Explanation / Answer

using energy conservation

Initial PE = final KE

KE = rotationalKE + translational KE = (Iw^2/2) + (mv^2 /2)

for solid cylinder ( I = mr^2/2)

mgh = ( mr^2/2 ( v/r)^2) /2   +   mv^2 /2

mgh = mv^2/4 + mv^2/2 = 3mv^2 /4

v = sqrt(4gh/3)

for hollow cylinder,

mgh = ( mr^2 ( v/r)^2) /2   +   mv^2 /2

mgh = mv^2/2 + mv^2/2 = mv^2

v = sqrt(gh)

vhollow / Vsolid = ( sqrt(gh) / sqrt(4gh/3)) = 0.866

B) rot KE of hollow = mv^2 /2

rot KE of solid = mv^2 /4

Khollow / Ksolid   = (mv^2/2) / (mv^2/4) = 2

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