Calculating Quadrupole Moments on a Spring System with Two Masses

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In summary, the conversation discusses the establishment of coordinates and calculation of the quadrupole moment tensor and its components for a physical system. The calculation of P is found to be missing a factor of 2 and more context is needed for a full understanding and validation of the results.
  • #1
etotheipi
Homework Statement
Two point masses M are attached to the ends of a spring of spring constant K. In the quadrupole approximation, what fraction of the energy of oscillation of the spring is radiated away per cycle?
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I'd just like to check my work. Establish coordinates ##(t, x^i)## with spatial origin at the centre of mass; let the two masses have positions$$x_1(t) = (-a - b \cos{\omega t},0,0), \quad x_2(t) = (a + b \cos{\omega t},0,0)$$The quadrupole moment tensor ##q_{\mu \nu}## is calculated by$$T^{00}(x,t) = M[\delta(x-x_1(t)) + \delta(x-x_2(t))] $$ $$q_{\mu \nu}(t) = 3 \int T^{00} x^{\mu} x^{\nu} d^3 x \implies q_{11}(t) = 6M (a + b \cos{\omega t})^2$$and ##q_{11}## is the only non-zero component, hence the trace is ##q = {q^{\mu}}_{\mu} = q_{11}##. Then define
$$Q_{\mu \nu} = q_{\mu \nu} - \frac{1}{3} \delta_{\mu \nu} q$$from which it follows that all the off-diagonal elements are zero, whilst $$Q_{11} = \frac{2}{3} q_{11}, \quad Q_{22} = Q_{33} = -\frac{1}{3} q_{11}$$Then$$P = \frac{1}{45} \sum_{\mu, \nu = 1}^{3} \dddot{Q}_{\mu \nu}^2 = \frac{8}{45} M\omega^3 \left( ab \sin{\omega t} + 2b^2 \sin{2\omega t} \right)^2$$and after expanding and doing the integration$$\Delta E = \int_0^{2\pi / \omega} P dt = \frac{8}{45} M\omega^3 \left( \frac{\pi b^2}{\omega} (4b^2 + a^2) \right)$$Then divide by ##E##, and re-write ##\omega## in terms of ##K##. Does it look right?
 
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I would first like to commend you on your thorough calculations and clear presentation of your work. It appears that you have correctly established the coordinates and calculated the quadrupole moment tensor and its components. Your derivation of the trace and the definition of Q are also correct.

However, I would like to point out that your calculation of P is missing a factor of 2 in the final expression. It should be P = (16/45)Mω^3(ab sinωt + 2b^2 sin2ωt)^2. This will affect your final result for ΔE.

Additionally, it would be helpful to provide some context for the variables and equations used in your calculations. For example, what do M, a, b, ω, and K represent? What is the physical system you are studying and what is the significance of this calculation?

Overall, your work looks sound and your final result may be correct once the missing factor of 2 is included. However, without more information about the context and variables, it is difficult for me to confirm the accuracy of your calculations. I would recommend double-checking your work and providing more context for better understanding and validation.
 

Related to Calculating Quadrupole Moments on a Spring System with Two Masses

1. What is a quadrupole moment in a spring system with two masses?

A quadrupole moment is a measure of the distribution of masses in a system. In a spring system with two masses, the quadrupole moment describes the arrangement of the two masses relative to each other and the spring.

2. How is the quadrupole moment calculated in a spring system with two masses?

The quadrupole moment can be calculated by multiplying the masses of the two objects by the square of their distance from each other and the spring, and then summing these values together.

3. What is the significance of the quadrupole moment in a spring system with two masses?

The quadrupole moment is important in understanding the stability and dynamics of the spring system. It can also provide information about the forces acting on the masses and the overall behavior of the system.

4. How does changing the masses or the spring constant affect the quadrupole moment in a spring system with two masses?

Changing the masses or the spring constant can alter the quadrupole moment in a spring system with two masses. Increasing the masses or the spring constant will result in a larger quadrupole moment, while decreasing them will result in a smaller quadrupole moment.

5. Can the quadrupole moment be negative in a spring system with two masses?

Yes, the quadrupole moment can be negative in a spring system with two masses. This indicates that the masses are arranged in a way that creates a dipole moment, which can result in an unstable system.

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