Solving for F to Hold a Box Still on a Ramp w/ Friction

[k-rod AP 2010]

Homework Statement

Write a General Formula solving for F to hold a box still on a ramp with friction.

there is no numbers in this, only variables

Homework Equations

F=ma

F=force m=mass a=acceleration θ=angle μ_s=static friction g=gravity

The Attempt at a Solution

i drew a free body diagram to separate the forces into components and plugged in the forces acting on the box to keep it from sliding of the frictioned ramp into F=ma

ma=Fcosθ₁ - [(μ_s*Fsinθ₁) + (m*g*sinθ₂)]

then, i made ma equal to 0 b/c the box isn't moving so a is 0

0=Fcosθ₁ - (μ_s*Fsinθ₁) + (m*g*sinθ₂)

This is the final equation i came up w/ after solving for F
F=(m*g*sinθ₂) / (cosθ₁ + μ_s*sinθ₁)

would this be correct?

 

[gau_physics]

Instead of making a=0 later its better you balance all the forces in horizontal and vertical direction since box is in equilibrium

There will be two conditions :
1) F is pushing force 2) F is pulling force

if condition (1), the Normal force increases thus increasing friction and just opposite in condition (2). You haven't mentioned what is θ2

 

[k-rod AP 2010]

theta2 is the angle the ramp is sloping at. theta1 is the angle the box is sloping at.

And do you mean have 1equation for horizontal forces and 1 equation for vertical? I am not sure what you mean