Question

A 7 m ladder weighing 250 N is being pushed by force F AT bottom What
is the minimum force needed to get the ladder to move? The static coefficient of friction for all contact surfaces is 0.4.

Answer 1

draw a digram. Notice on the wall, there is a horizontal force H1 into the wall, and a friction force on the wall downward, and some component of weight at vertically downward (V1)
At the base, there is a vertical force (v2), a component of weight, a horizontal pushing F, and a friction force opposing F.

Now you have several equilibrium equations you can write.
1) sum of vertical forces= 250
V1+V2=250
2) Sum of horizontal forces = zero
F-frictionbase- H1=0
3) Now write a moment equation about any point, I choose the center of the ladder. Assume the ladder makes an angle theta with the floor.
(F-frictionbase)7/2 SinTheta-(V2-frictionupwall)7/2 * cosTheta=0
Now, unknowns: friction base is a function of theta and V1. Frictionwallup is a function of H1, theta. V2 is a function of V1
So I see three equations, unknowns H2, V1, F so you should get an equation in terms perhaps of theta. I didn't work this out, but will be happy to check your work, but I am thinking it well be messy, so be careful.