Mass and spring in circular motion.

[AbigailM]

Just preparing for a physics prelim and working through previous exam questions.

Homework Statement

A mass m moving in a circular orbit about the origin is attracted by a three dimensional harmonic potential,

$U(r)=\frac{1}{2}kr^{2}$

What is the frequency of the orbit? If a small kick is supplied in the radial direction, what will be the frequency of the ensuing small oscillations in r?

The Attempt at a Solution

$k(r-r_{0})=m\omega^{2}_{0}r$

$\frac{k}{m}\frac{(r-r_{0})}{r}=\omega^{2}_{0}$

$\mathbf{Frequency\hspace{1 mm} of\hspace{1 mm} orbit\hspace{1 mm}}\omega_{0}=\sqrt{\frac{k}{m}\frac{(r-r_{0})}{r}}$

$\ddot{r}=-\omega^{2}r \hspace{5 mm} \omega^{2}=\frac{k}{m}$

$\omega^{2}=\frac{k}{m}=\frac{r\omega^{2}_{0}}{r-r_0}$

$\mathbf{Frequency\hspace{1 mm} of\hspace{1 mm} oscillation\hspace{1 mm}}\omega=\omega_0\sqrt{\frac{r}{r-r_{0}}}$

Just wondering if my solution looks correct. Thanks for the help.

 

[Infinitum]

Hi AbigailM

Your method looks correct to me, but you aren't given r₀ in the question...

 

[AbigailM]

Ok, so where I'm confused is that as the spring-mass is rotated, it extends from its equilibrium length due to centripetal force. So why is $(r-r_{0})$ wrong?

Thanks for the help.

 

[Infinitum]

Ok, so where I'm confused is that as the spring-mass is rotated, it extends from its equilibrium length due to centripetal force. So why is $(r-r_{0})$ wrong?

Thanks for the help.

It isn't wrong. The 'equilibrium length' r₀ isn't a given information in your question, you have assumed it(unless you wrote an incomplete question here :uhh:). And normally, we give answers in terms of stuff that's known to us. What do you think would be the value of r₀ in known terms?

 

[AbigailM]

Ahhh ok I see now. The restoring force is balanced by the centrifugal force from the rotation. We can set $r_{0}=0$. The reason I included $r_{0}$ is because for a harmonic oscillator, the mass oscillilates around the origin $r_{0}$. Any many cases it's defined to be zero. And I think your right, if we were to consider it in this problem it would have been given.

Thanks again for the help Infinitum.

 

[Infinitum]

Ahhh ok I see now. The restoring force is balanced by the centrifugal force from the rotation. We can set $r_{0}=0$. The reason I included $r_{0}$ is because for a harmonic oscillator, the mass oscillilates around the origin $r_{0}$. Any many cases it's defined to be zero. And I think your right, if we were to consider it in this problem it would have been given.

Thanks again for the help Infinitum.

Looks correct now