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Three Springs in an Equilateral Triangle

  • #1
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Homework Statement


[/B]
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)

Homework Equations


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!
 

Answers and Replies

  • #2
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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.
 
  • #3
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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
 
  • #4
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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?
 
  • #5
19,915
4,092
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
 

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