Mass-spring-mass system rotating about its centre of mass relativistically

[aayush.saxena]

Homework Statement

2 equal masses 'm' are attached to a spring of spring constant 'K' and unstretched length 'L'. The system is rotating about its centre of with an angular velocity 'w'. The linear velocity 'v' of m is comparable to that of speed of light 'c'. Find a relation between the Angular Momentum 'J' and the total energy 'E'.

Homework Equations

Hooke's law.
Centripetal force.
relativistic mass equation.

The Attempt at a Solution

I started off by equating the Force in the spring K (2r - L), where 2r is the compressed length of the spring; with the Centripetal force γm.w^2.r

I obtained the expression for r using this. Using r, i obtained the Angular momentum using J = (2γm).w.r^2

Now I cannot figure out how to obtain a relation between J and the total energy E.

Help? Thanks in advance.

 

[mark726]

To find a relation between the Angular Momentum J and the total energy E, we can use the following steps:

1. Start by writing the equation for the total energy of the system, which includes both the kinetic energy and potential energy. For a rotating system, the total energy can be expressed as:

E = KE + PE = 1/2mv^2 + 1/2K(2r-L)^2

2. Next, we can use the relativistic mass equation to express the mass m in terms of its rest mass mo and the velocity v. This gives us:

m = mo/√(1-(v/c)^2)

3. Now, we can substitute this expression for m into the equation for total energy, and also use the expression for r that you obtained earlier. This gives us:

E = 1/2(mo/√(1-(v/c)^2))v^2 + 1/2K(2(γm)w^2r^2-L)^2

4. Simplifying and rearranging, we get:

E = (mo/2)(v^2/√(1-(v/c)^2)) + K(γm)^2w^2(r^2-L)^2

5. Finally, we can use the expression for Angular Momentum that you obtained earlier, J = (2γm)w(r^2), and substitute it into the equation for total energy. This gives us the desired relation between J and E:

E = (mo/2)(v^2/√(1-(v/c)^2)) + KJ^2/(4γ^2m^2(r^2-L)^2)

This is the final relation between Angular Momentum J and total energy E for the given system.