# Mass of a picture in static equilibrium

## Homework Statement

An artist must push with a minimum of 75N at an angle of 45° to a picture to hold it in equilibrium. The coefficient of friction between the wall and the picture frame is 0.30. What is the mass of the picture?

ΣF=0
Ffr=μFN
F=ma

## The Attempt at a Solution

I set up an equilibrium equation of the y components of all the forces to find the mass. the y components would be, the force of gravity acting on the picture, the force of friction between the picture and the wall, and the y component of the applied force of the artist on the picture. Making the up forces equal to the down forces, you should get, Fappsin(45)=Fg-Ffr. Plugging in values into the known variables you should get, 75sin(45)=m(9.8)-(0.3)m(75sin(45). I factored out m and divided both sides by everything in the brackets to solve for m, obviously though I didn't get the answer.

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The problem is how the equation for the force of friction was used. The correct form is
$$F_{fr}=\mu F_{N}$$
where ##F_{N}## is the normal force between the picture and the wall.

The problem is how the equation for the force of friction was used. The correct form is
$$F_{fr}=\mu F_{N}$$
where ##F_{N}## is the normal force between the picture and the wall.
Yeah sorry, I realized the proper term would be FN, but it wouldn't matter because the normal force would still be 75sin(45) since the x component of 75N would be the normal force acting on the picture.

It matters because there is no mass term in the expression for ##F_{N}##.

It matters because there is no mass term in the expression for ##F_{N}##.
I'm confused, isn't the expression for FN=mass x gravity, where gravity is the x component of the applied force?

Gravity only operates in the y direction, so it cannot produce a force in the x direction. Here ##F_{N}## is the x component of the force applied by the person pushing the painting against the wall.

Gravity only operates in the y direction, so it cannot produce a force in the x direction. Here ##F_{N}## is the x component of the force applied by the person pushing the painting against the wall.
The x component is going to be still 75sin(45) since it's a 45 degree. I could use 75cos(45) but cos(45) and sin(45) are the same value it doesn't matter which trig function I use. I'm just baffled because everything seems right to me.

$$F_{N}=m*75\sin(45)$$
$$F_{N}=75\cos(45)$$
$$F_{N}=m*75\sin(45)$$
$$F_{N}=75\cos(45)$$