A wedge of angle θ rises towards the right and accelerates horizontally leftward with a time-dependent acceleration a(t)=βt. A block rests on its rough face. Given tanθ < μ < cotθ, the block initially sticks. The first time at which it starts slipping is:
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Question

A wedge of angle θ rises towards the right and accelerates horizontally leftward with a time-dependent acceleration a(t)=βt. A block rests on its rough face. Given tanθ < μ < cotθ, the block initially sticks. The first time at which it starts slipping is:

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G1 Physics · Accepted Answer

Because tanθ<μ, the block can initially remain at rest. As a increases, the required friction f = m(g sinθ-a cosθ) first decreases to zero at a=g tanθ, then reverses direction. The first limiting condition after reversal is a = g(sinθ+μcosθ)/(cosθ-μsinθ). Since a=βt, t_c = g(sinθ+μcosθ)/[β(cosθ-μsinθ)].