# Alpha Particle Scattering and angular momentum

Tags:
1. Feb 14, 2019 at 3:28 AM

### brotherbobby

Statement of the problem :

"Using the definition L = r $\times$ p, prove that the direction of L is constant for an alpha ($\alpha$) particle whose scattering is shown in the diagram below. "

Relevant equations :

We are aware that the scattering takes place via a central force F = F(r) $\hat r$. Angular momentum L = r $\times$ p and torque $\tau = r \times F$ (all vectors)

The attempt at a solution

I can solve the problem, but not the way it asks. The torque $\tau = r \times F \Rightarrow \tau = 0$ since the force is central : F = F(r) $\hat r$. Using $\tau = \frac{dL}{dt} = 0$, inplies that the angular momentum vector L is constant.

[This is not what the question asks for. It asks to show only the direction of L conserved, from the definition of L : L = r $\times$ p].

2. Feb 14, 2019 at 3:41 AM

### Orodruin

Staff Emeritus
If the vector itself is constant, then its direction must be conserved too ...

Also, where do you think $\vec \tau = \vec r \times \vec F$ comes from?

3. Feb 14, 2019 at 3:50 AM

### brotherbobby

Yes, in that sense I have answered the question, but as I said, not in the way they asked for. I suppose they want one to focus on the direction of L and show that it remains the same.
I don't know. Torque and angular momentum are definitions, defined as the cross product of the radius vector and the force or the linear momentum, respectively.

4. Feb 14, 2019 at 4:11 AM

### Orodruin

Staff Emeritus
So you need to relate those two, which you are going to do by computing the time derivative of the angular momentum.

5. Feb 14, 2019 at 4:40 AM

### brotherbobby

All I am aware of is the standard : $\frac{dL}{dt} = \frac{d}{dt}(r \times p) = r \times \frac{dp}{dt} = r \times F$.

It is this that I used to answer why the angular momentum L is constant - viz. the thrque is 0 owing to the fact that the force is central and hence L remains the same.

But the question asks me to show that the direction of L is a constant for a central force like this. (I am aware that I have already shown this in the form of the vector L itself being a constant).

I wonder if there is another way.

For instance, a vector L = $L\hat L$. Can we show that $\hat L$ is a constant from the definition of L ( = $r \times p$) for a central force (F(r))?