# Three Springs in an Equilateral Triangle

## Homework Statement

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Three identical point masses of mass m each are placed at the vertices of an equilateral triangle and joined through springs of equal length and spring constant k . The system is placed on a smooth table. If the masses are displaced a little towards the centroid of the triangle then time period of oscillation of the system is :
A)2π√(m/k)
B)2π√(m/2k)
C)2π/√(m/3k)
D)2π√(m/5k)

F = -k x
ω = √(k/m)

## The Attempt at a Solution

Consider one mass which is displaced from the mean position by x units. The two forces acting on it are k*x* cos(30 Degree) each inclined at an angle of 60 Degrees.

That would mean that the force acting on it is actually (3/2)*k*x, or the equivalent spring constant is (3/2)*k.

But that gives an weird time period which isn't in the options!

Bystander
Homework Helper
Gold Member
If the masses are displaced a little towards the centroid
"Masses," plural.
Consider one mass which
It's necessary to solve the stated problem rather than an intuitive simplification of the statement.

Chestermiller
Mentor
In what direction does the mass move?
If S is the side of the triangle, what is the radius of the circumscribed circle?
If S changes by ΔS, by how much does the radius of the circumscribed circle change?
What is the tension in each spring if S changes by ΔS.
What is the resultant F of the adjacent tension forces on the mass if S changes by ΔS?
How is the resultant force F on the mass related to its displacement?

Chet

1. The mass moves towards along the radius of the circumcircle.

2. The radius(R) of the circumcircle is S/√3

3. ΔR = ΔS/√3

4. T = k ΔS/√3

5. F = k ΔS

Could you please tell if these are correct?

Chestermiller
Mentor
1. The mass moves towards along the radius of the circumcircle.

2. The radius(R) of the circumcircle is S/√3

3. ΔR = ΔS/√3

4. T = k ΔS/√3

T = kΔS
5. F = k ΔS
F=2Tcos(30)=T√3

Now, combine these to express F in terms of ΔR.

Chet