Mechanics Marathon 5- a LoopHole!
A uniform solid spherical ball starts from rest on a loop-the-loop track . . It rolls without slipping along the track. However, it does not have enough speed to make it to the top of the loop. From what height h would the ball need to start in order to land at point P directly underneath the top of the loop? Express your answer in terms of R, the radius of the loop. Assume that the radius of the ball is very small compared to the radius of the loop, and that there are no energy losses due to friction.
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1 solution
In my opinion, the answer should be:
h = 1 0 2 7 R
The kinetic energy of a purely rolling sphere sphere is
K = 1 0 7 m v 2
Applying energy conservation between the highest and lowest point of the loop, we get:
K i n i t i a l = 2 m g R + K f i n a l
For the sphere to just reach the top, the reaction force at that point, on the sphere should be zero. Assuming the radius of the sphere is very small, leads to the equation:
R m v f i n a l 2 = m g
Solving the two equations gives:
v i n i t i a l 2 = 7 2 7 g R
The answer can be found by using the expression:
K i n i t i a l = m g h
Since none of the options point to this answer, it is possible that I may be wrong, in which case I request feedback from any other user. There is also a possibility that I have imagined this scenario incorrectly. It helps to have a diagram in the problem statement.