Statics of a Particle?

Question

A Peg-bag of mass m hanging from a washing line may be modelled as a particle suppored in equilibrium by two straight strings inclined at angles x and y to the horizontal, as shown in the diagram.
Find the tension in each string when: m=0.5, x=10 degrees and y=12 degrees

http://i174.photobucket.com/albums/w111/suned05/Tension.jpg

I Think you need to find the weight of the particle by doing 0.5*9.8=4.9 therefore the conbined tension of the 2 strings= 4.9, and using the sine or cosine rule find out how it splits but I have no idea- anyone?

Answer 1

→
F₁ = F₁ cos x î + F₁ sin x ĵ

→
F₂ = F₂ cos y î + F₂ sin y ĵ

→
W = -mg ĵ

Horizontal components:

∑ F = 0

F₁ cos x - F₂ cos y = 0

F₁ cos x = F₂ cos y

F₁ = F₂ (cos y/cos x)

Vertical components:

∑ F = 0

F₁ sin x + F₂ sin y - mg = 0

F₁ sin x + F₂ sin y = mg

(F₂ (cos y/cos x)) sinx + F₂ sin y = mg

F₂(cos y tan x + sin y) = mg

F₂ = mg/(cos y tan x + sin y) = 0.5 kg(9.8 m/s²)/(cos 12° tan 10° + sin 12°) = 4.9N/(0.3803) = 12.9N

F₁ = F₂ (cos y/cos x) = 12.9N (cos 12°/cos 10°) = 12.8N

Answer 2

Every body in equilibrium has a total sum of forces applied on it equal to zero. This rule is valid for separate directions, like this:
On the vertical axis the body has its weight applied on it, W=mg, plus the vertical parts of the tensions (facing up to counteract gravity, so that TotalForceVer=0 you'll see it for yourself). On the horizontal axis the body receives the horizontal parts of the tensions. Each part counteracts the other, so that TotalForceHor=0.
Assume positive directions (downwards-rightwards)
Let's find the parts of the tension of magnitude T1 on the left. Assume it points upwards
(it would be no mistake to take in any case the opposite direction -the final result would have the opposite sign, giving the right direction of the force).
It has an horizontal part -T1cosx and a vertical part -T1sinx.
The tension T2 on the right has
an horizontal part of T2cosy and a vertical part of -T2siny.
ΣFhor=0, so -T1cosx+ T2cosy=0 (Notice we use the algebraic forms).
Also, ΣFver=0, so mg-T1sinx-T2siny=0.
You can solve this system.