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

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

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 - μ_kN = ma_x

Step 6: Solve

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

Substitute into the x equation to give:

ma_x = Fcosθ - μ_k(mg - F sin θ) = F(μ_ksin θ + cosθ) - μ_kmg.

Maximize a_x. Compute da_x/dθ and set it to zero. This gives: tan θ_c = μ_k.

Finally, determine (a_x)_max by evaluating at the critical angle.