### Inclined Pulley

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.

1. Diagram and coordinate systems
2. Isolate the systems
3. Draw all forces acting
4. Take components
5. Apply F=ma and constraints
6. Solve
7. Check!

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:
ax =
 Mg sin θ - μk Mg cos θ - mg m + M

Mg sin θ - μk N - max - mg = Ma