How Do You Calculate the Force Needed to Prevent a Block from Sliding on a Wall?

[EventingLyn]

Homework Statement

A block of mass 7 kg is pushed against a wall by force P. The coefficient of static friction between the block and the wall is 0.42. Determine an expression for the force P as a function of θ such that the block will not slide up the wall or fall down the wall.
a.) Draw a free-body diagram for this problem
b.) Determine the force value needed if θ = 13°
c.) What is the minimum force required such that the block will remain stationary, and at what angle should this minimum force be applied?

Homework Equations

mg - f = Psinθ
n = Pcosθ
f = μ_scosθ

The Attempt at a Solution

I honestly wasn't sure where to start out since we haven't done any vertical force problems in class (I'm in a summer mini term, so my teacher tends to just rush through everything). So far, this is what I have, but I know that I am probably completely off.

After doing my FBD, I decided that I could do part C to find the minimum force needed, but I just kept the angle at 13°, so I'm not sure I did that correctly, either.

mg - μ_scosθ = Psinθ
P(sinθ + μ_scosθ) = mg
P = mg / sinθ + μ_scosθ
P = 7(9.81) / sin13° + (.42)(cos13°)
P = 108.28 NThank you so much for helping! All of us in the class are just drowning right now.

 

[haruspex]

but I just kept the angle at 13°,

Consequently you have found the answer to part b, at least on the assumption that the force direction is above the horizontal.
Along the way, there are a couple of typos. You missed a P in the first equation and you should use parentheses to make the order of operations right.
The very first part of the question is to find expressions in terms of a general theta for bounds on P such that the block will neither slide up nor slide down. How does that affect your general equation (before you plugged in 13 degrees)?

Part c requires you to find the least P over all possible angles.