Maximizing Fridge Stability: Determining the Optimal Height for Pushing Force

[faradayscat]

Homework Statement

say a fridge of width "w" and height "L" is being pushed on by a force F at an angle θ to the horizontal. This force is applied at a height of "h" above the ground.

I want to know what the max value of h can be such that the fridge doesn't tip and the coefficient of static friction is μ.

Homework Equations

ΣF = 0
Στ = 0

The Attempt at a Solution

The force can be found:

ΣFx = 0
Fcosθ - f = 0, where f(max) = μN

And

ΣFy = 0
N - mg - Fsinθ = 0

So

Fcosθ - μ(mg + Fsinθ) = 0
F = μmg/(cosθ - μsinθ)

This is my reasoning for the height, since only the horizontal component of F affects the perpendicular distance "h" to F, then:

Στ = 0 (about axis where the fridge is about to tip)
hFcosθ - mg(w/2) = 0
h = ½mgw/(Fcosθ)

Does that make sense? I'm skeptical about this.

 

[drvrm]

ΣFy = 0
N - mg - Fsinθ = 0

So

Fcosθ - μ(mg + Fsinθ) = 0
F = μmg/(cosθ - μsinθ)

How you are pushing? F acting below the horizontal at theta angle or above the horizontal?

 

[faradayscat]

How you are pushing? F acting below the horizontal at theta angle or above the horizontal?

Oups forgot to mention, the force is pushing above the horizontal, the vertical force is downward though

 

[drvrm]

If you are pushing in a direction above the horizontal then sin component will point up -in the direction of N

N - mg - Fsinθ = 0

so the sign of of F sin(theta) should change, if i am correctly following you!