I'm trying to model the velocity of a roller-coaster carriage as it goes down a circular hill.
To do this first I used energy equations: $$mgh_1+\frac{1}{2}mu^2=mgh_2+\frac{1}{2}mv^2 + E$$ (where E is the energy lost to friction and $\theta$ the angle of the carriage makes with the hill) $$v^2=u^2+2gr-2gr\cos(\theta)-2E/m$$ Then I need the energy lost to friction $$E=\frac{\int_0^\theta \lvert \mu mg\cos(\theta)-\mu mv^2/r \rvert d\theta}{\theta}2\pi r \theta /2\pi$$ $$E=r\int_0^\theta \lvert \mu mg\cos(\theta)-\mu mv^2/r \rvert d\theta $$
I can't integrate this though because its not all in terms of $\theta$. The v is changing. I tried again using force equations
$$a=\frac{\int_0^\theta(mg\sin\theta - \mu mg\cos(\theta)+ \mu mv^2/r)d\theta}{\theta} $$ (where a is the average acceleration)
But again there's little I can do here because there's a $v^2$ . Is there anyway I can get out of this loop? I feel like I'm going in circles.
Newton's Law.
Let $\theta=0$ be at the top of the sphere, $\theta=\pi/2$ be the far right and so falling straight down.
$mr^2\frac{d^2\theta}{dt^2}=-br^2\frac{d\theta}{dt}+mgr\sin\theta -mgr\mu \cos \theta$
Total Torque is equal to air resistance plus gravity plus kinetic friction. Will probably need a numerical method for that.
So this is non-linear.
This also assumes whatever is keeping the cart attached to the track cancels out gravity and centrifugal force.
That friction term at the right makes things complicated.
I don't think there is a closed form solution given its resemblance to the kinetic equation for pendulums.