Ungh 2. An inclined plane makes an angle of with the horizontal, as shown above.
ID: 3308163 • Letter: U
Question
Ungh 2. An inclined plane makes an angle of with the horizontal, as shown above. A solid sphere of radius R and mass M is initially at rest in the position shown, such that the lowest point of the sphere is a vertical height h above the base of the plane. The sphere is released and rolls down the plane without slipping. The moment of inertia of the sphere about an axis through its center is 2. Express your answers in terms of M, R. h, g, and . a. Determine the following for the sphere when it is at the bottom of the plane: i. Its translational kinetic energy ii. Its rotational kinetic energyExplanation / Answer
translationaL KE = m v^2 / 2
and rotational KE = I w^2 / 2
for rolling without slipping, w = v/ R
rot. Ke = (2 M R^2 / 5) (v/R)^2 / 2 = M v^2/ 5
Applying energy conservation,
PEi + KEi = PEf + KEF
M g h + 0 = 0 + (M v^2 / 2 + M v^2 / 5)
g h = 7 v^2 / 10
v^2 = 10 g h / 7
(i) translational KE = M v^2 / 2 = 5 M g h / 7
(ii) rot. KE = M v^2 / 5 = 2 M g h / 7
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