Newton's law in 2D

Question

A 1.5 kg book is held against the wall by an applied force. The applied force is 60° above the horizontal. Given that the static coefficient of friction between the block and the wall of 0.65, determine:

 

a) the maximum applied force required to hold the block in place

 

b) the minimum applied force required to hold the block in place

 

 

Answer 1

If the normal force is not zero, then equilibrium of forces in the horizontal direction gives

 

Fcosθ = N

 

N = normal force, which points horizontally

 

Equilibrium of forces in vertical direction gives

 

Fsinθ + μN = mg

 

(a)

 

The applied force is maximum when N = 0 ⇒

 

F_maxsinθ = mg

 

F_max = mg/sinθ = (1.5)(9.8)/sin60° newtons ≅ 16.97 newtons

 

(b)

 

The applied force is minimum when N ≠ 0.

 

N = Fcosθ

 

Fsinθ + μ(Fcosθ) = mg

 

F_min = mg/(sinθ + μcosθ) = (1.5)(9.8)/(sin60° + 0.65cos60°) newtons ≅ 12.34 newtons