A block with mass m is being pushed by a constant force F that makes an angle of θ with the horizontal

A block with mass m is being pushed by a constant force F that makes an angle of θ with the horizontal as shown below. The block is moving with constant velocity on a level surface. The coefficient of kinetic friction between the block and the surface is µk. Which one of the following equations is correct for the magnitude of F?

Solution

Let |F| is the magnitude of the force F

Given Data: m, F, θ, a=0, µ_k

                      |F| = |F(m, θ, µ_k)|=?

F_NET=0 because a=0

X direction:   |F|·cos θ + f = F_{NET, X} = 0

Y direction: N – mg –  |F|·sin θ = F_{NET, Y} = 0,

where N is the normal force,
             g is the gravity acceleration

f= µ_k N

|F|·cos θ + µ_k (mg- |F|·sin θ) = 0

|F|·cos θ + µ_k mg - µ_k |F|·sin θ = 0

|F|·cos θ  - µ_k |F|·sin θ = µ_k mg

|F|·( cos θ  - µ_k sin θ) = µ_k mg

|F| = µ_k mg/( cos θ  - µ_k sin θ)