# Is angular momentum conserved only for the center point of the centripetal force?

From my book,

Consider a centripetal force is applying on a object:

$$\vec{F}=f(r)\hat{r}$$

The torque for a particular point is,

$$\vec{N}=\vec{r}\times\vec{F}=\vec{r}\times{f(r)\hat{r}}$$

$$\vec{N}=r\hat{r}\times{f(r)\hat{r}}$$

$$\vec{N}=rf(r)(\hat{r}\times\hat{r})=\vec{0}$$

So the angular momentum is conserved for any centripetal force.

I tried to figure out this using gravitational force that is applied to a satellite in orbit from earth.

I firstly selected the origin as the center of the earth, in this view point force is constant and it is anti-parallel to the position vector of the satellite. So the torque of the satellite always $$\vec{N}=\vec{0}$$, the angular momentum of the satellite is conserved.

Then I selected another origin that is random. For this origin, the position vector is anti-parallel to the gravitational force vector on the satellite for one situation. For all other situations the position vector and the force vector are not anti-parallel so the torque of the satellite with respect to the non-center of centripetal force origin is $$\vec{N}\ne\vec{0}$$.

So, Is the angular momentum conserved only when we choose the origin as the origin of the centripetal force?

So, Is the angular momentum conserved only when we choose the origin as the origin of the centripetal force?

Linear momentum is conserved when there are no external forces.

Angular momentum is conserved when there are no external torques.

By selecting an axis that that is not coincident with the earth, you've created a situation where the earth's force creates a torque on the satellite (when considered from this new axis). So yes, in this situation we would not expect the angular momentum of the satellite to be conserved.

When we set up problems, we will often consider angular momentum about a particular axis because it simplifies the problem. Both calculations are valid, but we have to know more information and do more work when we pick an axis that is not coincident with the earth's center of mass in this case.

• Hmm, I get it. So, the "the angular momentum is conserved for any centripetal force." quote is valid when we choose the origin as the center of the centripetal force. Am I correct? Commented Dec 24, 2019 at 18:46
• Yes. Angular momentum is conserved when there are no external torques. When all forces are directed to or from the origin, there are no torques. Commented Dec 24, 2019 at 18:49
• Alright. Thanks for your time. Have a good day! Commented Dec 24, 2019 at 18:54

In the way you've framed the problem, yes.

However, if you had two bodies, of mass $$m_{1,2}$$ and positions $$\mathbf r_{1,2}$$, defining the relative position $$\mathbf r=\mathbf r_2-\mathbf r_1$$ with a force

$$\mathbf F=f(r)\hat r$$

acting between them, angular momentum would be conserved w.r.t. any point.