Mass of a picture in static equilibrium

[Zack K]

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?

Homework Equations

ΣF=0
F_fr=μF_N
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, F_appsin(45)=Fg-F_fr. 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.

 

[NFuller]

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.

 

[Zack K]

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 F_N, 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.

 

[NFuller]

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

 

[Zack K]

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

I'm confused, isn't the expression for F_N=mass x gravity, where gravity is the x component of the applied force?

 

[NFuller]

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.

 

[Zack K]

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.

 

[NFuller]

Your work says
$$F_{N}=m*75\sin(45)$$
There is no mass term. It should be
$$F_{N}=75\cos(45)$$

 

[Zack K]

Your work says
$$F_{N}=m*75\sin(45)$$
There is no mass term. It should be
$$F_{N}=75\cos(45)$$

Oh wow it just hit me, haha I feel so dumb. Since the force has already been given by 75sin(45), a mass term would be ridiculous.