Two climbers are on a mountain. Simon, of mass m, is sitting on a snow covered slope that makes an angle θ with the horizontal. The coefficient of static friction between his body and the snow is μ. He is tied into one end of a massless rope that runs over a frictionless pulley. Joe, of mass M, is at the other end of the rope. He has fallen and is hanging motionless below an overhang. Derive an expression for the maximum value of Joe's mass M so that Simon is not pulled down the slope, in terms of relevant system parameters.

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Physics

Craig Meyer

19 April, 19:58

Two climbers are on a mountain. Simon, of mass m, is sitting on a snow covered slope that makes an angle θ with the horizontal. The coefficient of static friction between his body and the snow is μ. He is tied into one end of a massless rope that runs over a frictionless pulley. Joe, of mass M, is at the other end of the rope. He has fallen and is hanging motionless below an overhang. Derive an expression for the maximum value of Joe's mass M so that Simon is not pulled down the slope, in terms of relevant system parameters.

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Ty Cochran · Answer 1

It is required that the weight of Joe must prevent Simon from being pulled down. That means he is not slipping down but tends to be towed down. So in equilibrium, force of friction will act in upward direction on Simon.

Let in equilibrium, tension in rope be T

For balancing Joe

T = M g

For balancing Simon

friction + T = mgsinθ

μmgcosθ+T = mgsinθ

μmgcosθ+Mg = mgsinθ

M = (msinθ - μmcosθ)

M = m (sinθ - μcosθ)