A solid sphere is released from height h from the top of an incline making an an

Question

A solid sphere is released from height h from the top of an incline making an angle ? with the horizontal. (a) Calculate the speed of the sphere when it reaches the bottom of the incline in the case that it rolls without slipping. (Use any variable or symbol stated above along with the following as necessary: g for the acceleration of gravity.) vf = (b) Calculate the speed of the sphere when it reaches the bottom of the incline in the case that it slides frictionlessly without rolling. (Use any variable or symbol stated above along with the following as necessary: g for the acceleration of gravity.) vf = (c) Compare the time intervals required to reach the bottom in cases (a) and (b). rolling time/sliding time =

Explanation / Answer

Use conservation of mechanical energy in both parts (K + U)i = (K + U)f

Set U f = 0 at the bottom and Ki = 0 at the top (at rest)

a) Ui = m*g*h & K f = 1/2*m*v^2 + 1/2*I*^2 (this term accounts for the rotational K)
Now
I for a solid sphere = 2/5*m*R^2 and
= v/R so

K = 1/2*m*v^2 + 1/2*2/5*m*R^2*v^2/R^2

So K = 1/2*m*v^2(1+2/5) = 0.7m*v^2

So 0.7m*v^2 = m*g*h so v = (g*h/0.700)

b) Now K = 1/2*m*v^2 Setting equal to Ui we get 1/2*m*v^2 = m*g*h so v = (2*g*h)

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