Physics Question

Question

A metal block with mass m = 10 kg is connected by a cord wrapped around a 1.5-kg hollow cylinder pulley with a frictionless axle. The surface of the incline has a coefficient of friction μk = 0.60. As the block slides down the inclined plane from rest at an angle of 30 degrees, it pulls the cord and causes the pulley to rotate. If the inclined plane has a length of 3m, calculate the speed v of the block before it reaches the bottom. The pulley has a moment of inertia of a solid disk, I = mr^2/2 , and a radius r = 0.25 m.

Answer 1

The net force on the block is mgsinθ - μmgcoθ - F_T = ma (F_T is the force of tension of the rope) We have picked down the incline as positive. The parallel component of weight - μ x perpendicular component of weight is typical from the incline geometry.

The net torque on the cylinder is F_TR = Iα ( The rotational analog of Newton's Second Law) Substituting 1/2 MR² for I and a/R for α

F_TR = 1/2 MR²(a/R) or F_T = 1/2 Ma

Combine the equations by eliminating F_T and solve for a

mgsinθ - μmgcosθ - 1/2 Ma = ma

a =( mgsinθ - μmgcosθ )/(1/2M +m)

Now that a is known (just plug in data), you can calculate v from

2ad = v² - v₀² and solve for v.