what is the minimum angle of a smooth inclined plane so that a body kept on top takes least time to reach bottom of plane ?
Let's consider a smooth ramp in horizontal position, with a block kept at one end.
If we start to incline the ramp slowly keeping the other end fixed, the block will start moving down the incline.
It's accleration is gsinθ.
Hence the length, L of the ramp will be traversed, in
t = (L/gsinθ)
since, L = gsinθ t2/2......as the initial velocity is zero.
Hence the minimum time corresponds to maximum value of sinθ which is one and the angle is 90.
But we need the minimum angle, well then if we take the angle less than 90 degrees time will not be least.
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