My real life problem is to calculate initial translational and angular velocities of a vehicle in a loss of control to a stop. (The vehicle will translate and rotate about it's center of mass.)
Initial strategy is to use energy-work theorem, therefore:
$K = W$
where $K$ is the initial kinetic energy and $W$ is the work due to friction (F) between the tires and the road surface (assume that's the only external force at play), therefore:
$\frac 1 2 mv^2 + \frac 1 2 I\omega^2 = \int F\,ds + \int \tau\,d\theta$
Assume I know how to calculete the RHS of the equation. The problem with this, is that i have two variables($v$ and $\omega$) with only one equation.
The question is: Is it valid to write two different equations, one for rotation and the othe for translation, as following?
$\frac 1 2 mv^2 = \int F\,ds$
$\frac 1 2 I\omega^2 =\int \tau\,d\theta$
This way i would be able to solve for $v$ and $\omega$, but I am not so sure i can do it without violating some underliying principle..