My question concerns these components of a question addressing the motion of a particle expressed in polar:
With that, here's the particular part of the question I'm trying to answer:
From my knowledge (which will surely have flaws in it due to the fact that I wrote this question), the angular momentum can be expressed using linear momentum from the following thought process:
$$\vec L = \vec r \times\vec p$$
$$\vec L = \vec r \times m \vec v$$
Which, I guess, answers the first bit?
Now, onto the next one.
I'm not sure how this can be true along with the vector expression simultaneously, but I know this is true as well:
$$L = rmv_{\theta}$$
And $$v_{\theta} = \frac{d\theta}{dt}$$
And recalling what $\theta$ is defined as in terms of time in the first part of the question -
$$\dot \theta = \beta t^2$$ $$r = \frac{\alpha}{t}$$
Thus, $$L = m\frac{\alpha}{t} \beta t^2$$
This is clearly not the result I'm trying to confirm. I think it has to about something with the $\hat z$ unit vector, which I'm not sure about what it is or what it's doing there, and that I'm using $v_{\theta}$, not $\vec v$, which is the vector sum $v=v_r + v_{\theta}$. Can someone explain to me what I ought to consider and do next here?