- #1

- 128

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## Main Question or Discussion Point

Consider a central force. The central force is radial by definition, so ##\vec{F}=f(r) \hat{r}##. Therefore, by definition, the acceleration caused by the force, in the direction of ##\hat{\theta}## must be zero, ##\vec{a_{\theta}}=0##.

In presence of central force angular momentum is conserved, in particular its magnitude, which is ##|\vec{L}|=m r^2 v_{\theta}##, where ##v_{\theta}## is the magnitude of the velocity of the body in the direction of ##\hat{\theta}##.

That means that the product ##r^2 v_{\theta}## stays constant and, if ##r## gets smaller, ##v_{\theta}## must increase.

Since ##v_{\theta}## is the magnitude of the velocity of the body in the direction of ##\hat{\theta}##, this means that there must be and acceleration in this direction, necessarily.

But how can this be possible, since, as said, the force is completely radial?

I'm ok with the fact that the force is not orthogonal to the velocity, hence the magnitude of the velocity can increase, but, in particular, I really don't see how the magnitude of the component ##v_{\theta}## can change, provided the fact that the force is radial.

In this picture it is clear that the magnitude of the velocity (which is tangential to the ellipse, of course) changes, but I can't understand the change in ##v_{\theta}##.

View attachment 99029

In presence of central force angular momentum is conserved, in particular its magnitude, which is ##|\vec{L}|=m r^2 v_{\theta}##, where ##v_{\theta}## is the magnitude of the velocity of the body in the direction of ##\hat{\theta}##.

That means that the product ##r^2 v_{\theta}## stays constant and, if ##r## gets smaller, ##v_{\theta}## must increase.

Since ##v_{\theta}## is the magnitude of the velocity of the body in the direction of ##\hat{\theta}##, this means that there must be and acceleration in this direction, necessarily.

But how can this be possible, since, as said, the force is completely radial?

I'm ok with the fact that the force is not orthogonal to the velocity, hence the magnitude of the velocity can increase, but, in particular, I really don't see how the magnitude of the component ##v_{\theta}## can change, provided the fact that the force is radial.

In this picture it is clear that the magnitude of the velocity (which is tangential to the ellipse, of course) changes, but I can't understand the change in ##v_{\theta}##.

View attachment 99029