Combination of 3 springs at given angles

[watermlon]

Three springs of spring constant kare connected to a body of mass m and rigid supports, so that the angles are
A 90 degrees between two and 135 degrees between each of these two and the third.
B 120 degrees between any 2 adjacent ones.
Find time period of vibration.
I worked out a method to solve this question. I added 2 of the spring constants vectorially, first dividing each by the cos of half of the angle between them (45 and 60 degrees) and the third simply as k to get a net spring constant. This gave a correct answer but I don't remember what logic I used and it may not have been correct. Could someone show me the correct solution?
diagram
http://i898.photobucket.com/albums/ac182/watermlon/df.jpg

 

[tiny-tim]

Welcome to PF!

I added 2 of the spring constants vectorially, first dividing each by the cos of half of the angle between them (45 and 60 degrees) and the third simply as k to get a net spring constant. This gave a correct answer but I don't remember what logic I used and it may not have been correct.

Hi watermlon! Welcome to PF!

What matters is the components of force in each direction.

If the extension vectors are x y and z (these are not coordinate directions), then the forces are k₁x + k₂y + k₃z …

in this case, k₁ = k₂ = k₃ = k,

so the total force is k(x + y + z), and you correctly took components along an axis of symmetry (the total of components along the perpendicular axis would be zero).

 

[watermlon]

Thanks, tim. I understand it now.