Homework Help: Torque and angular momentum

1. Mar 26, 2016

erisedk

1. The problem statement, all variables and given/known data
The torque $\vec{τ}$ on a body about a given point is found to be equal to $\vec{A} × \vec{L}$ where $\vec{A}$ is a constant vector, and $\vec{L}$ is the angular momentum of the body about that point. From this it follows: (Multiple answers correct)

(A) $\dfrac{d\vec{L}}{dt}$ is perpendicular to $\vec{L}$ at all instants of time

(B) the component of $\vec{L}$ in the direction of $\vec{A}$ does not change with time

(C) the magnitude of $\vec{L}$ does not change with time

(D) $\vec{L}$ does not change with time

2. Relevant equations
$\vec{τ} = \dfrac{d\vec{L}}{dt}$

3. The attempt at a solution
$\vec{τ} = \dfrac{d\vec{L}}{dt} = \vec{A} × \vec{L}$
From this equation (A) holds.

(D) will hold, i.e. only if $\dfrac{d\vec{L}}{dt}$ is 0, i.e.$\vec{A}$ is parallel to $\vec{L}$ which has no reason to be true all the time. So, D should not be correct.

Which leaves (B) and (C). I have no idea how to prove or disprove them. Please help.

2. Mar 26, 2016

Orodruin

Staff Emeritus
Why don't you simply try finding some expressions for the vectors relevant to B and C and differentiate them with respect to time?

3. Mar 26, 2016

erisedk

Thank you! Got it. I differentiated these two expressions:
For (B)
$\vec{L}.\vec{L} = L^2$
and for (C)
$\vec{L}.\vec{A} / A$

4. Mar 26, 2016

erisedk

Hence, (A) (B) and (C) are true.

5. Mar 26, 2016

Orodruin

Staff Emeritus
Correct.