More physics help (Friction)?

Question

A block of mass m rests on a rough horizontal surface. Coefficient of friction is u. A pulling force F = mg acts at angle θ with the vertical slide of the block. Find when the block can be pulled along the floor.

(a) Cot θ ≥ u

(b) tan θ ≥ u

What are the limits of cot θ in terms of u if the force is pushing rather than pulling?

If θ is smaller than a certain angle α, the block cannot be moved no matter what the force. Find α in terms of arccot u

Answer 1

F makes θ with vertical. Therefore, horizontal component of F = Fsinθ and vertical component = Fcosθ

After breaking F into these two components, we find that the forces on the block are

1. Fsinθ forward = mgsinθ forward

2. Fcosθ upward = mgcosθ upward (since F = mg)

3. Friction f backward

4. Normal force N upward

5. Weight mg downward

There is no acceleration in vertical direction. Therefore, total force in vertical direction = 0

Or, mgcosθ + N-mg = 0

Or, N = mg-mgcosθ

Or, N = mg(1-cosθ)

f = uN = umg(1-cosθ)

Block can be pulled when horizontal component of F exceeds friction.

Or, mgsinθ > umg(1-cosθ)

Divide by mg

sinθ > u(1-cosθ)

Do we have to choose between a and b? Or are a and b two different questions?