Can someone help me work this out please? Thanks! You have a pair of inclined pl

Question

Can someone help me work this out please? Thanks!

You have a pair of inclined planes such that a block that slides down one can slide up

the other without losing any energy in the transition. The inclined planes are both at an

angle ? from the horizontal, as shown in the diagram. The inclined plane on the left is

frictionless. You start the block sliding down either side from a height h. Which side do

you start the block sliding down to have it slide the greatest distance up the other side?

a) right side

b) left side

c) it doesn

Explanation / Answer

Let:
s1, s2 be the distances travelled along the first and second planes,
m be the mass,
g be the acceleration due to gravity,
a be the angle of inclination of each plane to the horizontal.
u be the coefficient of friction.

Starting on the smooth plane:
mg s2 sin(a) = mg s1 sin(a) - umg s2 cos(a)
s2 = s1 - u s2 cot(a)
s2 = s1 / [ 1 + u cot(a) ].

Starting on the rough plane:
mg s1 sin(a) - u mg s1 cos(a) = mg s2 sin(a)
s1 - u s1 cot(a) = s2
s2 = s1 [ 1 - u cot(a) ].

The smooth plane first is preferable if:
1 / (1 + u cot(a) ] > 1 - u cot(a)
1 > 1 - [ u cot(a) ]^2
That's always.

b) left side

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