Pulling a Trunk

You are pulling a heavy trunk of mass m along a level floor in which the coefficient of sliding is μk. You can pull with a force of fixed magnitude F.

Question:  What direction should you pull to accelerate as quickly as possible?

Answer:  DID TASC!
  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!

Steps 1-4: See transparency

Step 5: Apply Newton's 2nd law

y-component: F sin θ + N = mg    x-component: F cosθ - μkN = max

Step 6: Solve

The y equation gives: N = mg - F sin θ.

Substitute into the x equation to give:

max = Fcosθ - μk(mg - F sin θ) = F(μksin θ + cosθ) - μkmg.

Maximize ax. Compute dax/dθ and set it to zero. This gives: tan θc = μk.

Finally, determine (ax)max by evaluating at the critical angle.