Finding Tension using Newton's Laws

[Mangoes]

Homework Statement

A picture frame hung against a wall is suspended by two wires attached to its upper corners. If the two wires make the same angle with the vertical, what must this angle be if the tension in each wire is equal to 0.75 of the weight of the frame. Ignore any friction between the wall and picture frame.

Homework Equations

Newton's laws of motion.

The Attempt at a Solution

The picture frame has a weight w.

The question states that the tension of the wire is 0.75w. Since the picture frame has zero acceleration, it is in equilibrium. I drew a picture to clarify my thought process:

http://i.imgur.com/oR8HST0.png

ƩF_x = 0

ƩF_x = T₂cos(θ) - T₁cos(θ) = 0
This tells us that T₂ = T₁ = 0.75w

ƩF_y = 0

ƩF_y = T₁sin(θ) + T₂sin(θ) - w = 0

ƩF_y = T₁sin(θ) + T₁sin(θ) - w = 0

= 2Tsin(θ) - w = 0

= 1.5wsin(θ) = w

= sin(θ) = (2/3)

Taking the arcsine gives approximately 42 degrees, but the angle is 48 degrees and I can't figure out what I'm doing wrong.

 

[Doc Al]

Taking the arcsine gives approximately 42 degrees, but the angle is 48 degrees and I can't figure out what I'm doing wrong.

I suspect that they want the angle with respect to the vertical, not the horizontal.

 

[tiny-tim]

Hi Mangoes!

… If the two wires make the same angle with the vertical, what must this angle be …

Nice diagram!

But you've marked the wrong angle as θ! ​

 

[Mangoes]

Oh wow...

I even typed it and everything and still didn't pick up that vertical part in my head.

Thanks a lot guys.

 

[SteamKing]

Usually, most pictures are hung using a single nail, so that the wires form an inverted V with respect to the picture.