Determining possible trajectories (Classical Mechanics)

[MrCreamer]

Homework Statement

Determine possible trajectories for particle with constant magnitude of velocity |$\dot{\vec{r}}$| = v₀ and constant angular momentum $\vec{L}$ = $\vec{L}$₀

Homework Equations

|$\dot{\vec{r}}$| = v₀
$\vec{L}$ = $\vec{L}$₀

The Attempt at a Solution

I know that L dot is zero and thereby the torque is zero. My intuition tells me that the possibly trajectories would be circles but mathematically, I am not sure where to start.

 

[BvU]

Look at he definition of L. Perhaps the time derivative of that tells you something

 

[MrCreamer]

Well, as I said, if $\dot{\vec{L}}$ = 0, then the net torque and net force are zero. The constant velocity magnitude implies that the direction of the velocity vector can change but the speed would remain constant.

If F is zero, then you have

m$\ddot{x}$ = 0

Which implies that x(t) is of some linear form:

x(t) = at + b, where a = v$_{0}$ and b = x$_{0}$.

I'm assuming the question requires the graphing of the trajectories in phase space and hence would require some form of mathematical development in terms of the energy of the system.

 

[vela]

I think BvU meant for you to differentiate $\vec{L} = \vec{r}\times\vec{F}$ and to interpret the result.

By the way, your inference that $\dot{\vec{L}} = 0$ implies $\vec{F}=0$ is not correct.

 

[BvU]

I meant: differentiate $\vec L = \vec r \times \vec p$ wrt time, knowing $\vec p = m\dot{\vec r}$. Don't venture into x and y because you have everything you need in polar coordinates.