A problem I am trying to work out is as follows:
A particle moves in a force field given by $\vec F =\phi(r) \vec r$. Prove that the angular momentum of the particle about the origin is constant.
I set it up as follows:
$\vec F = m {d^2\vec r \over dt^2}$
$\vec v = \int {\frac {\vec F}{m} }\ dt = \int {\frac {\phi(r) \vec r}{m} }\ dt$
which is equal to :
${\frac {\phi(r) t \vec r}{m} } + c$
(I am not sure what I am doing at this point. Is my integrated expression correct?)
Assuming it is, we get:
Angular Momentum $L = m (\vec r \times \vec v) = \vec r \times (\phi(r) t \vec r + c)$
Now I don't know what to do with the constant term, but I do know that
$\vec r \times k\vec r = 0$
However, the problem states that we have to prove the result is a constant, so I think I'm wrong. Specific places where someone could help me out are:
(1) Is my integration correct? If not, how does one integrate a force (given in terms of position vector notation) w.r.t. time?
(2) What happens to the constant? Cross-product of a vector and a scalar doesn't make any sense.