# How to solve this System of Second Order Differential Equations (DE) with IVP

• Jun 15th 2012, 09:14 PM
kashyappatel7
How to solve this System of Second Order Differential Equations (DE) with IVP
Hello, I'm trying to figure out how solve the below for varying time and/or corresponding theta. I'm stumped.. any insight is greatly appreciated. Thank you

$\displaystyle \ddot{R} = −\alpha|v|\dot{R} - \beta_{s}\frac{|v|}{R}\dot{R}-\beta_{c}\frac{g}{|v|}\dot{R} + [R\dot{\theta}^2-\frac{5}{7}g(tan\delta cos\varepsilon -sin\varepsilon cos\theta]cos^2\delta$

$\displaystyle \ddot{\theta} = −\alpha|v|\dot{\theta} - \beta_{s}\frac{|v|}{R}\dot{\theta}-\beta_{c}\frac{g}{|v|}\dot{\theta}- 2\dot{R}\dot{\theta}/R - \frac{5}{7}\frac{g}{R}sin\varepsilon sin\theta$

with $\displaystyle |v|=\sqrt{\dot{R}^2/cos^2\delta + R^2\dot{\theta}^2}$

where
$\displaystyle \alpha = 0.00900350489819847$
$\displaystyle \beta_{s} = 0.0022919958783044724$
$\displaystyle \beta{c}=0.0062192039382357378$
$\displaystyle \delta=0.2574$
$\displaystyle R=0.242$
$\displaystyle g=9.807$

with the 'initial conditions' at t = 0:

$\displaystyle R(0) = R$
$\displaystyle \dot{R}(0) = 0$
$\displaystyle \theta(0) = 57.899552605659885$
$\displaystyle \dot{\theta}(0) = 2.7371516764119637$

This is modeling a ball rolling inside of a cone with the tip of the cone facing down. So the ball starts off at the rim of the cone and spins around inside the cone going further and further towards the tip. A good example of this is to think about a roulete ball which is spinning around the rim of the roulette table and then slowly approachs the spinning numbers in the middle. As the ball slows down, gravity takes over and it approachs the center (tip) of the cone faster and faster.