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 ). 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