Understanding Torque and Angular Momentum Conservation

[erisedk]

Homework Statement

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

Homework Equations

$\vec{τ} = \dfrac{d\vec{L}}{dt}$

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.

 

[Orodruin]

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

 

[erisedk]

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

 

[erisedk]

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

 

[Orodruin]

Correct.