What is the Kinetic Energy of a Rotating Square Mass-Spring System?

[proximaankit]

Homework Statement

Four masses M in deep space are connected by four identical light springs with spring constant k and equilibrium length L. The four mass, four spring assembly is square and lies in a plane; all four masses are rotating with ω =√(k/M), in uniform circular motion about an axis perpendicular to the plane and equidistant from all four masses. What is the kinetic energy of this system?

Homework Equations

ma_c=ω²r
F=kx
KE=1/2*I_Totalω²

The Attempt at a Solution

I_Total=4MR² where R is the distance to mass when the masses are rotating. R should be large that L because the springs will stretch when the system is in motion.
R=(√2)/2(l+Δl)
I_Total=2*M*(l+Δl)²
KE=2*(l+Δl)²*k
I feel like this is incomplete especially with respect to Δl. Is there any other way to put it. I was thinking of maybe finding what Δl maybe using Hooke's Law but I don't know how to go about that. I was thinking of equating it to the centripetal force. Please let me know and thank you in advance

 

[Doc Al]

I was thinking of maybe finding what Δl maybe using Hooke's Law but I don't know how to go about that. I was thinking of equating it to the centripetal force.

You're on the right track.

Combine Newton's 2nd law with Hooke's law. Draw a diagram of the forces acting on each mass.

 

[gneill]

Homework Statement

Four masses M in deep space are connected by four identical light springs with spring constant k and equilibrium length L. The four mass, four spring assembly is square and lies in a plane; all four masses are rotating with ω =√(k/M), in uniform circular motion about an axis perpendicular to the plane and equidistant from all four masses. What is the kinetic energy of this system?

Homework Equations

ma_c=ω²r
F=kx
KE=1/2*I_Totalω²

The Attempt at a Solution

I_Total=4MR² where R is the distance to mass when the masses are rotating. R should be large that L because the springs will stretch when the system is in motion.
R=(√2)/2(l+Δl)
I_Total=2*M*(l+Δl)²
KE=2*(l+Δl)²*k
I feel like this is incomplete especially with respect to Δl. Is there any other way to put it. I was thinking of maybe finding what Δl maybe using Hooke's Law but I don't know how to go about that. I was thinking of equating it to the centripetal force. Please let me know and thank you in advance

Yes, use the centripetal force. The springs meet at each corner of the square and their tensions add vector-wise to make up the centripetal force. You can find an expression for the radial distance of the masses that way.