# Find the frequency of oscillation

1.
This question was in my test recently, and it bothered me I could not solve this.
The system is as follows:
There is a mass which hangs about 3 springs. 2 at each side and one connected to the top. The whole system is balanced, which makes the springs perpendicular to the mass. And each spring has the constant K.
Now we move the mass in the upwards direction, and so I need to find the frequency of oscillation.

2.
Ok. So this is what I did. I found the equilibrium point, that is when the force of the spring and the gravitational pull are the same, which that point will be my "center".
So when we get the mass upwards, it displaces a distance of Y from the center. The upper string exerts -KY on the mass. The other two springs at the sides are also contributing in the Y direction by (lets say the length of one of the springs when at rest is X1)
{sqrt( (X1)^2 + Y^2) - X1} ---- this part is to figure out how much it displaced from its resting position.
now I transform it to the Y direction by multiplying it by sin(theta) which is
Y/ {sqrt( (X1)^2 + Y^2) - X1}.
All in all Ive got forces working on the mass as
F= MG -KY + KY(1 - X1/{sqrt( (X1)^2 + Y^2) - X1}) + KY (1 - X2/{sqrt( (X2)^2 + Y^2) - X1}).
I dont think I was wrong with my analysis of the forces, but now i need to get those Y's in the square root out, so i can work out the frequency, But I dont know how. Which means I probably did something wrong :)

If anyone read all this and was not deterred by the length of the question, PLZ help :)

It seems all right to me. But it is possible that the question had to be solved assuming x1 to be quite long so that sin(theta) tends 0. in such case only vertical spring is important.

It is just a guess as I do not know the level of difficulty in your exams.