Can Resistance Determine the Torque in a Solenoid Pendulum System?

[jore1]

Homework Statement

A solenoid of length b is made of a heavy wire of density p and radius r. It is tightly wound around a cylinder of radius a and suspended at either end from light, conducting wires of length l from a fixed horizontal conducting rod (see diagram below). When the solenoid is set swinging in a vertical magnetic field B, it oscillates at a slightly different frequency compared to a simple pendulum of the same mass and length. Calculate this frequency shift.

Relevant Equations

ε = -dΦ/dt

The flux enclosed by the loop consisting of the solenoid, wires and conducting rod at an angle θ is Φ = blBsinθ, then using small angle approximations and differentiating the induced emf can be found.

I know that there must be some torque opposing the motion but am unsure how to proceed.

 

[kuruman]

There is the usual torque due to gravity and a magnetic torque $\vec \tau_M=\vec \mu\times \vec B$ where $\vec \mu$ is the magnetic moment of the closed loop through which the induced current runs.

 

[Fred Wright]

Following the suggestion of @kuruman, you may observe that the magnetic moment in the coil produced by the induced current wants to align itself in the direction of the external magnetic field, i.e. it will try to twist the coil in the direction of the suspending wires. Will this change the natural frequency of the pendulum?

 

[kuruman]

Following the suggestion of @kuruman, you may observe that the magnetic moment in the coil produced by the induced current wants to align itself in the direction of the external magnetic field, i.e. it will try to twist the coil in the direction of the suspending wires. Will this change the natural frequency of the pendulum?

You are suggesting a torsional mode. My suggestion was to consider the retarding torque $\vec \mu\times \vec B$ where the magnitude of the magnetic moment is the induced current multiplied by the area $l\times b$ of the rectangular current loop.

 

[jore1]

You are suggesting a torsional mode. My suggestion was to consider the retarding torque $\vec \mu\times \vec B$ where the magnitude of the magnetic moment is the induced current multiplied by the area $l\times b$ of the rectangular current loop.

Thank you for your reply
Would the magnetic moment of the solenoid be included and would the induced current be found though the self inductance of the loop and solenoid?

 

[kuruman]

Thank you for your reply
Would the magnetic moment of the solenoid be included and would the induced current be found though the self inductance of the loop and solenoid?

The magnetic moment of the solenoid would affect the torsional mode and itself inductance would affect the induced current. My sense is that you are expected to produce an approximate answer by writing the full blown differential equation for the angular acceleration, making the necessary approximations and deducing the frequency from the diff. eq. without actually solving it. You are given enough information to calculate the mass and inductance of the solenoid but you are not given the relevant resistance that will enable you to find the induced current and hence the magnetic moments. I suppose you can call it $R$ and proceed.

Also, it would make more sense if the problem stated "When the solenoid is set swinging in a vertical magnetic field B, it oscillates at a slightly different frequency compared to the oscillation frequency when the field is off." This way one compares apples with apples because we are given a physical pendulum and are asked to compare it with a simple pendulum.

 

[jore1]

The magnetic moment of the solenoid would affect the torsional mode and itself inductance would affect the induced current. My sense is that you are expected to produce an approximate answer by writing the full blown differential equation for the angular acceleration, making the necessary approximations and deducing the frequency from the diff. eq. without actually solving it. You are given enough information to calculate the mass and inductance of the solenoid but you are not given the relevant resistance that will enable you to find the induced current and hence the magnetic moments. I suppose you can call it $R$ and proceed.

Also, it would make more sense if the problem stated "When the solenoid is set swinging in a vertical magnetic field B, it oscillates at a slightly different frequency compared to the oscillation frequency when the field is off." This way one compares apples with apples because we are given a physical pendulum and are asked to compare it with a simple pendulum.

Thank you for the reply. I now better understand the problem.

 

[rude man]

A good problem. My thoughts:
There are three conceivable emf generators:
(i) the coil loops forming the solenoid
(ii) the paraxial wire connecting the coils to each other
(iii) the suspension wires.

I would focus on only one of these.

As @kurumaan pointed out, resistance R is needed.
Now you can compute emf and current in the circuit.

The rest is determing the source of torque from one or more of the above three sections. If you're handy with 2nd order damped oscillator equations you can go to them, or solve the ODE.