2) Prove than tan ϑ is equal to uk (coefficient of kinetic friction) when a block slides down an incline with a constant speed. (use symbols, not numbers)
If the ϑ is the maximum angle of incline just before the block moves, what is us (coefficient of static friction) in terms of ϑ? The block slides with a constant speed down the incline because the net force on it is zero. Thus, the component of its weight acting downward along the plane i.e. mg sinθ is balanced by the force of friction fs = μ k mg cosθ. where mg cosθ is the normal reaction Thus, mg sinθ = μ k mg cosθ, or μ k = tanθ. If a block is placed on an incline, and the angle of inclination is gradually increased, at a particular angle, the block just starts moving. Thus the force of static friction opposes the component of weight down the incline till this inclination is reached and this is the max value of the force of static friction. Here also the above equations apply ; there is no motion because there is no net force, but the coefficient of friction used id the coefficient of static friction. mg sinθ = μ s mg cosθ, or μ s = tanθ.s |