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

(a) in this situation sphere have translational and rotational kinetic energy

from conservation of energy at bottom and top

0.5mv^2 +0.5Iw^2 = mgh

moment of inertia of solid sphere I = (2/5) mr^2

for without slipping condition

v = rw

0.5mv^2 +(0.5)(2/5)mv^2 = mgh

v = (10gh/7)^0.5

(b) in this case it have only translation kinetic energy

from conservation of energy

0.5 mv^2 = mgh

v = (2gh)^0.5

(c) time = distance/velocity

t1/t2 =v2/v1

t1/t2 = (7*2gh/10gh)^0.5

t1/t2 = 1.18

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