I was asked to identify a constant of motion which does not depend on the mass of the object whilst in contact with the surface. enter image description here

I found the equation of motion of the material point but I don't know how to identify/find a constant of motion independent of mass? How should I approach the problem?

equations of motion projected on axes: enter image description here

the answer for "find a constant of motion which does not depend of mass while the point is in contact with the surface of the cone" is the following but I don't understand what it represents: enter image description here

  • $\begingroup$ what does r^2 times dphi/dt represent? $\endgroup$ – SoHCahToha Jan 19 '17 at 11:37
  • $\begingroup$ I don't know what "cste" & "selon" are. $\endgroup$ – JMac Jan 19 '17 at 11:40
  • $\begingroup$ cste means constant and "selon" is french for "with respect to " $\endgroup$ – SoHCahToha Jan 19 '17 at 11:54

The final equation of motion corresponds to the conservation of angular momentum. So $\dot{phi}$ is a first integral. Do you know how to solve the last equation for the first integral? Divide by $m$ and $sin\theta$ and then you can integrate directly to obtain $h=r^2 \dot{\phi}$ where $h$ is a constant of motion. Basically what you are looking for is $\frac{d}{dt}$(some expression)$=0$. Then when you integrate with respect to time you will have a constant on the RHS.

This was tough to write on an iPhone. Good luck!

  • $\begingroup$ Thanks! but how do you integrate it once you divide by msintheta? $\endgroup$ – SoHCahToha Jan 19 '17 at 12:07
  • 1
    $\begingroup$ Can you get to $\ddot{\phi}/\dot{\phi} + 2\dot{r}/r = 0$? $\endgroup$ – Rumplestillskin Jan 19 '17 at 12:26
  • 1
    $\begingroup$ Once you integrate this you'll get $\ln(\dot{\phi}) + 2\ln(r) = \ln(h)$ where $h$ is a constant. Which is equal to $r^2 \dot{\phi}. = h$. $\endgroup$ – Rumplestillskin Jan 19 '17 at 12:29
  • 1
    $\begingroup$ thank you very much! I now know how to look for a constant of motion( look for expressions such that d/dt = 0 and integrate it to get the expression = const ! :) $\endgroup$ – SoHCahToha Jan 19 '17 at 12:33
  • $\begingroup$ How do you know that the final equation corresponds to the conservation of angular momentum ? the answer is rangularvelocity (m^2 /s) should it be rangularvelocity*m to be the angular momentum? Ahhh or is it because we divided by m at the start ? $\endgroup$ – SoHCahToha Jan 19 '17 at 16:41

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.