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.
Step 1: Diagram and coordinate system.
Define +x = down the ramp and +y = perpendicular to the ramp.
Steps 24: 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 xcomponent 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:
a_{x}  = 

Mg sin θ  μ_{k} N  ma_{x}  mg = Ma
Step 7: Check the answer.
What happens for θ=0? θ=π?