Question

Hi there, I've got a calculus question. The moment of angular momentum M = r*ma = r*m(dv/dt)
This equation is then expanded out to:
d/dt(r*mv) - (dr/dt *mv) = d/dt(r*mv) - (v*mv)
I'm not sure where the subtracted term comes from in the first step, reminds me a bit of the quotient rule, nor where the v in the subtracted term comes from in the second. Any explanation, or a nudge in the direction of where my knowledge is insufficient would be greatly appreciated. Thanks!

Answer 1

this is a bit verbose, especially the switching between ways of saying the same thing, ie \(\dfrac{ d \vec r}{dt} = \vec v \) and \(\dfrac{ d \vec v}{dt} = \vec a \)

i reckon you realise that the fact you need is the application of the product rule to the vector cross product:

\(\dfrac{d}{dt} ( \vec r \times m \vec v ) = \vec v \times m\vec v + \vec r \times m\vec a\)

re-arrange this to get

\(\vec r \times m \vec a = \dfrac{d}{dt} ( \vec r \times m\vec v) - \vec v \times m\vec v \)

and the punchline is.....

\(\vec v \times m\vec v = m|\vec v|^2 sin 0 = 0 \)


so

\( \vec r \times m\vec a = \dfrac{d}{dt} ( \vec r \times m\vec v) \)

the term Moment of Angular Momentum is not one i have heard but Angular Momentum is \(\vec L = \vec r \times \vec p = \vec r \times m\vec v\), and i suppose that is the Moment of Linear Momentum

hope that helps...