2
$\begingroup$

Consider a particle moving an a straight line, with constant velocity $v$. The angular momentum (pivot point $O$) can be calculated as $$L=mr v_{\theta}$$Where $v_{\theta}$ is the velocity perpendicular to the vector $r$ at each istant. enter image description here

Now if I calculate the angular momentum in $A$ it get $L_A=mr_A v_x$, while in $B$ I get $L_B=mr_B v_y$.

In general $L_A\neq L_B$ but how can that be? How can angular momentum not be conserved? There are no forces, or torques!

I'm probably missing something big but I cannot see the mistake

$\endgroup$
1
  • $\begingroup$ What is special about your point O? Why would you think that anything to do with O should have any impact on some random particle sailing by? $\endgroup$
    – Jon Custer
    Apr 18, 2016 at 18:18

1 Answer 1

2
$\begingroup$

If you draw similar triangles, then you'll find that $r_A/r_B = v_y/v_x$, and so the product $r_A v_x$ is equal to $r_B v_y$. Try drawing a line from the tip of your lower $\vec{v}$ vector to the tip of your lower $v_y$ component to see this.

$\endgroup$
3
  • $\begingroup$ and these two ratios are simply the slope of the straight line $\endgroup$
    – Tofi
    Apr 18, 2016 at 18:24
  • $\begingroup$ @MichaelSeifert Thanks for the reply! If I may ask, so the fact that $L$ is conserved does not imply that $v_{\theta}$ (i.e. the velocity perpendicular to $r$) is constant, right? $\endgroup$
    – Sørën
    Apr 18, 2016 at 21:53
  • 1
    $\begingroup$ @Sørën: Only if $r$ is constant as well. This is the case, for example, for a planet orbiting the sun in a circle (which has a constant tangential speed at all times). On the other hand, a comet orbiting the sun in an elliptical orbit has a varying $r$ and a varying $v_\theta$, but it has a constant angular momentum $L$. (This is why the comet is moving faster when it's closer to the Sun.) $\endgroup$ Apr 18, 2016 at 22:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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