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## Homework Statement

A box with a mass of 22kg is at rest on a ramp inclined at 45 degrees to the horizontal. The coefficient of friction between the box and the ramp are mu static = 0.78 mu kinetic = 0.65.

a) Determine the magnitude of the largest force that can be applied upward, parallel to the ramp, if the box is to remain at rest.

b) Determine the magnitude of the smallest force that can be applied onto the top of the box, perpendicular to the ramp, if the box is to remain at rest.

## Homework Equations

Fa = ma

Fs = Fn mu static

Fk = Fk mu kinetic

## The Attempt at a Solution

Given:

m = 22kg

Incline angle = 45 degrees

mu static = 0.78

mu kinetic = 0.65

a)

The box is at rest, so Fg downward parallel to ramp = Fs upward parallel to ramp.

The largest force that can be applied upward parallel to ramp must be > Fs to accelerate from 0 m/s to some velocity to move.

But if I push the box from a lower altitude (upward) then would Fs be amplified by Fg parallel to ramp? Or does Fs already accommodate for gravity?

Fs = Fn mu static

Does Fn = Fg? Since there is no vertical movement? Does Fg = mg? Or is it modified by the inclination of ramp? Would it be Fg = mg cos 45?

Assuming Fg = mg sin 45

Assuming Fg = Fn = mg sin 45 = (22kg)(9.8N/kg)(cos45) = 152.46N

Fs = Fn mu static

Fs = (152.46N)(0.78) = 118.91N

Assuming Fs has gravity parallel to ramp from up to down taken into consideration:

Fa must be greater than 118.91N upward parallel to hill to accelerate the box.

b)

I'm not sure how to do this. Adding a force perpendicular to ramp is like increasing the Fg perpendicular to hill, hence increasing the Fn, which affects Fs.

Remain at rest needs all forces to be balanced, so Fg parallel to ramp = Fs parallel to ramp, Fg perpendicular to ramp = Fn.

I don't see how changing Fg perpendicular to ramp can move the box, unless it becomes greater than the stability of the surface of the hill and crushes it so the box collapses into the hill? :/

Fs = Fn mu static

Fn = Fg cos 45

Fg = mg

So the m is being questioned here.

If Fgx parallel to ramp is > Fs then it will move, so Fgx = Fs

So mg sin45 = mg cos45 mu static. But it doesn't.

Why is the Fs less than the Fgx but the object is still at rest?

I'm so confused at this point I don't think I've done anything right since the beginning.

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