A block of mass M= 2 kg lies on a 30 degree inclined plane and is connected to a mass m=0.2 kg by a string that passes over a pulley at the top of the incline. The friction coefficients for the block on the plane are: μs = 0.4 and μk = 0.3. Starting at rest, will the masses accelerate? If so, which direction?
Use DID TASC. Subtle feature: we don't know whether the masses will accelerate, so it's not immediately clear which friction force applies.
Step 1: Diagram and coordinate system.
Define +x = down the ramp and +y = perpendicular to the ramp.
Steps 2-4: See tranparency and diagram.
Step 5: Apply Newton's 2nd Law for mass M.
y direction: N - Mg cos θ = 0 or
N = Mg cos θ = 16.97 N
The maximum static friction force is fs max = μs N = 0.4 x 16.97 = 6.8 N.
x direction: Mg sin θ - fk - T = M ax (1)
Static friction (6.8 N) and tension in the rope (1.96 N) are always less than the x-component of the force of gravity (9.8 N). The block accelerates downhill, so there is a kinetic friction force acting up the slope.
Step 5 (again): Apply Newton's 2nd Law for mass m.
y direction: T - mg = may or T = may + mg (2)
Constraint: ax (for mass M)
equals ay (for mass m). Thus write ax= ay=a.
Step 6: Solve.
Substitute (2) into (1) gives:
Mg sin θ - μk N - max - mg = Ma
Step 7: Check the answer.
What happens for θ=0? θ=π?