Force Acting on Rope: Homework Solutions

[intdx]

Homework Statement

From http://library.thinkquest.org/10796/index.html (#6b, 6c)

A 10 kg picture is hanging on a wall by two ropes.
http://library.thinkquest.org/10796/ch5/IMG00008.GIF
...
b. Are the forces acting on the ropes A and B equal in magnitude?
c. How big is the force acting on the rope A?

The site accepts the answer 49N for #6c, but I'm not sure why.

Homework Equations

$$F_g=mg$$
$$F=\sqrt{F_x^2+F_y^2}$$
$$F_x=F\times \cos \theta$$

The Attempt at a Solution

The answer for #6c appears to be equal to
$$10\textrm{kg}\times 9.80\textrm{m/s}^2\times \textrm{cos }60^\circ=49\textrm{N,}$$
but that doesn't make sense to me; shouldn't we combine the weight of the picture with the rightward pull? In that case, how do we find the rightward pull?

Or, is there an imaginary diagonal force--equal in magnitude to the gravitational force--of which we want the horizontal component? In that case, why?

The answer to #6b is "No," but I'm not sure why. (I'm guessing that once I understand #6c, then #6b will follow.)

Thank you!

 

[intdx]

Actually, I suppose that combining the force of gravity in this case (98N) with a horizontal vector couldn't reduce the magnitude to just 49N...
Is it the second option that I proposed, then? $$\sqrt{F_g^2\ \textrm{cos}^2\theta +F_g^2\ \textrm{cos}^2(90^\circ-\theta)}=F_g\textrm{,}$$ so it seems plausible. The forces on the ropes, not including from the wall or ceiling where they're above, should add up to the force exerted by the picture, right?

Still, even if I've found the way to do the problem, I don't feel like I have a sufficient understanding of it.

 

[azizlwl]

You can resolve mg to 2 components of opposing direction to A and B since they are 90° apart.
To me it is mgCos30°.

 

[intdx]

You can resolve mg to 2 components of opposing direction to A and B since they are 90° apart.
To me it is mgCos30°.

I'm sorry; I don't understand what you mean. Would you mind drawing it?

Would it be something like this?

$$mg\textrm{cos(30}^\circ \textrm{)}$$

EDIT - whoops, sorry, I edited without realizing you had posted again.

 

[azizlwl]

I'm sorry; I don't understand what you mean. Would you mind drawing it?

Just extend line A and B downward. These 2 directions can be components of mg.

 

[Yukoel]

Hello intdx,
Try Lami's theorem here.Or otherwise resolve the weight along OA and OB to check for equilibrium.
regards
Yukoel