# Find r(dot)

• Oct 8th 2011, 02:06 PM
mezy
Find r(dot)
Can anyone show me how to find r(dot) of the following vector?

r = <x,y> + (lm)/(m+n)<cos(theta),sin(theta)>
• Oct 8th 2011, 02:51 PM
Plato
Re: Find r(dot)
Quote:

Originally Posted by mezy
Can anyone show me how to find r(dot) of the following vector?
$\displaystyle r = <x,y> + (lm)/(m+n)<\cos(\theta),\sin(\theta)>$

What does r(dot) mean?
In the definition of r what are the variables?
What are the constants?
• Oct 9th 2011, 05:53 AM
mezy
Re: Find r(dot)
Quote:

Originally Posted by Plato
What does r(dot) mean?
In the definition of r what are the variables?
What are the constants?

Sorry about that. Here is the complete question:

Two point masses $\displaystyle m_{1}$ and $\displaystyle m_{2}$ are joint together by a rigid light rod of lenth $\displaystyle \ell$. If the rod move on a vertical plane under the action of the earth's gravitational field only, show that the path of the center of mass is a parabola and the rod rotates about the center of mass at a uniform angular velocity.

The given information I have is:

$\displaystyle \vec{r_{1}} = <x,y> + \displaystyle{\frac{\ell m_{1}}{m_{1}+m_{2}}} <cos (\theta),sin (\theta)>$

and

$\displaystyle \vec{r_{2}} = <x,y> - \displaystyle{\frac{\ell m_{2}}{m_{1}+{m_{2}}} <cos (\theta),sin (\theta)>$

Prove that $\displaystyle \theta$ is cyclic.

To do this, the first step is to find $\displaystyle \dot{\vec{r_{1}}}$ and $\displaystyle \dot{\vec{r_{2}}}$ and use them to find the Euler-Lagrangian equation.

I hope that's a little more specific. I just need help finding $\displaystyle \dot{\vec{r_{1}}}$.

Thanks!
• Oct 20th 2011, 06:13 PM
Hartlw
Re: Find r(dot)
Quote:

Originally Posted by mezy
Sorry about that. Here is the complete question:

Two point masses $\displaystyle m_{1}$ and $\displaystyle m_{2}$ are joint together by a rigid light rod of lenth $\displaystyle \ell$. If the rod move on a vertical plane under the action of the earth's gravitational field only, show that the path of the center of mass is a parabola and the rod rotates about the center of mass at a uniform angular velocity.

The given information I have is:

$\displaystyle \vec{r_{1}} = <x,y> + \displaystyle{\frac{\ell m_{1}}{m_{1}+m_{2}}} <cos (\theta),sin (\theta)>$

and

$\displaystyle \vec{r_{2}} = <x,y> - \displaystyle{\frac{\ell m_{2}}{m_{1}+{m_{2}}} <cos (\theta),sin (\theta)>$

Prove that $\displaystyle \theta$ is cyclic.

To do this, the first step is to find $\displaystyle \dot{\vec{r_{1}}}$ and $\displaystyle \dot{\vec{r_{2}}}$ and use them to find the Euler-Lagrangian equation.

I hope that's a little more specific. I just need help finding $\displaystyle \dot{\vec{r_{1}}}$.

Thanks!

r = <x,y> +k<cosw,sinw>
dr/dt = <dx/dt,dy/dt> +k<-sinw(dw/dt),cosw(dw/dt)>
= <dx/dt,dy/dt> +k(dw/dt)<-sinw,cosw)>

assuming k constant. otherwise you have to differentiate that too. d/dt(k<x,y>=dk/dt<x,y> +k<dx/dt,dy/dt>