# Thread: Polar Coordinates - Motion

1. ## Polar Coordinates - Motion

A particle P, moving in a plane, has polar coordinates $\displaystyle (r , \theta )$ referred to the fixed pole O and the initial line in the plane

Given that the particle moves under a force directed towards towards O, show that at time t $\displaystyle r^2 \frac{d \theta}{dt} = h$.

Where h is a constant
I keep going around in circles with this. a little hint would be nice.

Many Thanks

Bobak

2. Originally Posted by bobak
I keep going around in circles with this. a little hint would be nice.

Many Thanks

Bobak
Consider the equations of motion:
$\displaystyle m \frac{d^2x}{dt^2} = a_x$
and
$\displaystyle m \frac{d^2y}{dt^2} = a_y$

What do these equations look like in polar coordinates?
(Hint: The solution is
$\displaystyle m \left [ \frac{d^2r}{dt^2} - r \left ( \frac{d \theta}{dt} \right )^2 \right ] = f_r$
and
$\displaystyle m \left [ 2\frac{dr}{dt} \frac{d \theta}{dt} + r \frac{d^2 \theta}{dt^2} \right ] = 0$
Make sure you know how to derive these.)

Or alternately if you are in an advanced enough class, do this using Lagrangian methods. It's easier.

-Dan

3. Originally Posted by topsquark
Consider the equations of motion:
$\displaystyle m \frac{d^2x}{dt^2} = a_x$
and
$\displaystyle m \frac{d^2y}{dt^2} = a_y$

As the force is directed towards the origin that does that mean that $\displaystyle a_x \propto -x$ and similarly for $\displaystyle a_y$ or not ?

What do these equations look like in polar coordinates?
(Hint: The solution is
$\displaystyle m \left [ \frac{d^2r}{dt^2} - r \left ( \frac{d \theta}{dt} \right )^2 \right ] = f_r$
and
$\displaystyle m \left [ 2\frac{dr}{dt} \frac{d \theta}{dt} + r \frac{d^2 \theta}{dt^2} \right ] = 0$
Make sure you know how to derive these.)
This is all new to me, I can't find anything about this in my books, would you mind pointing me to some reading material so I can learn how these results are derived?

Many Thanks Dan

Bobak