- #1
Soren4
- 127
- 2
I don't understand why the conservation of angular momentum can imply an acceleration, in absence of a force.
Consider for istance planetary motion. The angular momentum \vec{L} of the planets is conserved and that means \mid \vec{L} \mid=mr^2 \dot{\theta}=mrv_{\theta} is conserved too.
Consider the acceleration in polar coordinates
$$
\left( \ddot r - r\dot\theta^2 \right) \hat{\mathbf r} + \left( r\ddot\theta+ 2\dot r \dot\theta\right) \hat{\boldsymbol{\theta}} \ $$
The second term is zero since \vec{L} is constant. In fact the second term can be rewritten as <br /> a_{\theta}=\frac{1}{r}[\frac{d}{dt}(r^2 \dot{\theta})]=\frac{1}{r}[\frac{d}{dt}(\frac{L}{m})].
This means that there is no acceleration in the direction of \hat{\boldsymbol{\theta}}, which is clear since the gravitational force is a central froce.
But if the distance r decreases v_{\theta} (i.e. the velocity in the direction of \hat{\boldsymbol{\theta}}) must increase in order to keep \mid\vec{L} \mid constant.
How can v_{\theta} increase if there is no acceleration in the direction of \hat{\boldsymbol{\theta}}?
I understood that it happens because of the conservation of angular momentum but if there is an acceleration, necessarily a force is needed. I don't see where do this force come from.
Consider for istance planetary motion. The angular momentum \vec{L} of the planets is conserved and that means \mid \vec{L} \mid=mr^2 \dot{\theta}=mrv_{\theta} is conserved too.
Consider the acceleration in polar coordinates
$$
\left( \ddot r - r\dot\theta^2 \right) \hat{\mathbf r} + \left( r\ddot\theta+ 2\dot r \dot\theta\right) \hat{\boldsymbol{\theta}} \ $$
The second term is zero since \vec{L} is constant. In fact the second term can be rewritten as <br /> a_{\theta}=\frac{1}{r}[\frac{d}{dt}(r^2 \dot{\theta})]=\frac{1}{r}[\frac{d}{dt}(\frac{L}{m})].
This means that there is no acceleration in the direction of \hat{\boldsymbol{\theta}}, which is clear since the gravitational force is a central froce.
But if the distance r decreases v_{\theta} (i.e. the velocity in the direction of \hat{\boldsymbol{\theta}}) must increase in order to keep \mid\vec{L} \mid constant.
How can v_{\theta} increase if there is no acceleration in the direction of \hat{\boldsymbol{\theta}}?
I understood that it happens because of the conservation of angular momentum but if there is an acceleration, necessarily a force is needed. I don't see where do this force come from.