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: μ_s = 0.4 and μ_k = 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 f_{s max} = μ_s N = 0.4 x 16.97 = 6.8 N.

x direction: Mg sin θ - f_k - T = M a_x   (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 = ma_y    or    T = ma_y + mg   (2)

Constraint: a_x (for mass M) equals a_y (for mass m). Thus write a_x= a_y=a.

Step 6: Solve.

Substitute (2) into (1) gives:

ax

=

Mg sin θ - μk Mg cos θ - mg m + M

Mg sin θ - μ_k N - ma_x - mg = Ma

Step 7: Check the answer.

What happens for θ=0? θ=π?