6
$\begingroup$

Q: A mass $m$ hangs from a massless spring connected to the roof of a box of mass $M$. When the box is held stationary, the mass-spring system oscillates vertically with angular frequency $\omega$. If the box is dropped falls freely under gravity, why does the angular frequency increase?

So initially, we have $\omega=\sqrt{\frac{k}{m}}$, where $k$ is the spring constant. I read that the new angular frequency when in free-fall is $\omega'=\sqrt{\frac{k}{\mu}}$, where $\mu=\frac{Mm}{M+m}$ is the reduced mass, i.e. $\omega'=\sqrt{\frac{k(M+m)}{Mm}}$ which is clearly greater than $\omega$.

I'm not sure why this is true - I suspect it has something to do with the box oscillating (as it's no longer held stationary), but I'm not too familiar with the concept of reduced mass so I'd really appreciate a good explanation.

$\endgroup$
  • $\begingroup$ Are you looking for a more intuitive or a more mathematical explanation? $\endgroup$ – K. Kirilov Jan 13 at 21:40
  • $\begingroup$ I’d like a concrete mathematical one if possible $\endgroup$ – Max Jan 13 at 21:41
  • $\begingroup$ Is wikipedia's derivation enough? en.wikipedia.org/wiki/Reduced_mass#Newtonian_mechanics $\endgroup$ – K. Kirilov Jan 13 at 21:51
  • $\begingroup$ I understand the derivation, but I’m not sure how exactly it’s applied in this situation. Clearly two objects are the box and the mass $m$, but why is relative acceleration relevant to an observer looking on from outside the box? Also how does the spring affect this? $\endgroup$ – Max Jan 14 at 2:31
  • $\begingroup$ The following is more of an educated guess than a factual claim. If you think I'm in the right direction I could read up and try to provide an answer. In reality you always have $\omega = \sqrt{\frac{k(M + m)}{Mm}}$ But if $M$ is very large, compared to $m$ you have $M + m \approx M$ so the $(M + m)$ from the numerator kind of cancels out the $M$ in the denominator, so you're left with only $\sqrt{\frac{k}{m}}$. While the box is stationary $M$ is the mass of the box + the mass of the earth, so it is very large, compared to $m$ and when it is in free fall it is just the mass of the box. $\endgroup$ – K. Kirilov Jan 14 at 3:34
1
$\begingroup$

Despite the weird geometry of the problem, this is simply two masses connected by a spring, and we can solve that problem using Lagrangian Mechanics. Let's assume that the masses have mass $M$ and m, and position coordinates $x_M$ and $x_m$ respectivly. They are also separated by a distance $d$, and the spring constant is $k$.

We will use $\alpha$ as the generalized coordinate representing how far the masses have stretched from equilibrium: $$ \alpha = x_M - x_m - d $$

Note also that $$ \dot{\alpha} = \dot{x}_M - \dot{x}_m, $$ because $d$ is a constant, and we can choose a frame in which the total momentum of the system is zero, which gives us $$ m \dot{x}_m = -M \dot{x}_M $$

We can write the kinetic energy as $$ T = \frac{1}{2} m \dot{x}_m^2 + \frac{1}{2} M \dot{x}_M^2, $$ and then transform to our generalized coordinate $\alpha$ using the above two equations: $$ T = \frac{1}{2} m \dot{x}_m^2 + \frac{1}{2} M \Big(\dot{x}_m \frac{m}{M}\Big)^2 $$ $$ T = \frac{1}{2} m \Big( -\dot{\alpha} \frac{M}{m+M}\Big)^2 + \frac{1}{2} M \Big( \Big( -\dot{\alpha} \frac{M}{m+M}\Big) \frac{m}{M}\Big)^2 $$ $$ T= \frac{1}{2} \frac{m M^2}{(m+M)^2} \dot{\alpha}^2+ \frac{1}{2} \frac{m^2 M}{(m+M)^2}\dot{\alpha}^2 $$ $$ T= \frac{1}{2} \frac{m M}{m+M} \dot{\alpha}^2 $$ $$ T= \frac{1}{2} \mu \dot{\alpha}^2, $$ where $\mu$ is the reduced mass.

The potential energy is much easier to find: $$ U = \frac{1}{2} k \alpha^2. $$ Our Lagrangian then is $$ L = T-U $$ $$ L = \frac{1}{2}\mu \dot{\alpha}^2 - \frac{1}{2} k \alpha^2, $$ which can be plugged into the Euler-Lagrange equation

$$ \frac{d}{dt} \frac{\partial L}{\partial \dot{\alpha}} - \frac{\partial L}{\partial \alpha}=0. $$

This equation is easy to solve by hand: $$ \mu \ddot{\alpha}+k \alpha = 0. $$

This is clearly the equation for a simple harmonic oscillator with angular frequency $\sqrt{k/\mu}$, which is the answer you were looking for!

$\endgroup$
0
$\begingroup$

A mass hangs by a spring from the roof of a box under gravity, and the mass-spring system oscillates vertically with angular frequency w.

Then the box is dropped and falls freely while the mass is moving downward.

The string is stretched until the box and the mass are traveling at the same velocity. Then the spring pulls the mass and the box toward each other. There is no force to push them apart until they hit each other, and the problem does not say how elastic their collision is.

So the problem does not give the information needed to tell whether they will continue to oscillate or what the frequency will be if so.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.