Trajectory of a rolling ball with uneven weight distribution A perfect ball is rolling on a plane. Without further forces, it would roll in a straight line, and that's it.
What, however, if the ball's weight distribution is uneven? For example, the ball might have a higher-density smaller ball placed within it, but being slightly off-center, so the center of gravity is not the geometric centroid of the ball. Note that I assume the ball to be solid, i.e., no moving parts within the ball.
How would this ball roll then? Which trajectories would be realizable by such a ball?
Sadly, I don't know enough mechanics to even write down the differential equations and simulate it on a computer...
 A: Well, as the center of mass is not coinciding with the normal force(vertically upwards), there will be a net torque about the point during the motion of the ball. When the heavier ball is towards the front, it will support rolling and increase the rotational speed of the ball. When it will go to back, it will give the opposite effect. As net energy is conserved, the ball will slip(considering no friction) if it started with pure rolling. You can write the equations as follow, just consider the motion of the center of the heavier ball. Let it's distance from the center of the main ball be r. Now, x will be equal to $r\cos \theta $,where theta is the angle from the horizontal axis passing through the center of the main ball. This will give you the torque as (M-m)gx where m is the mass of the equivalent part of the other side. This will give you $\tau=I\dot{w}$. Now, solve for the w where w is $d\theta/dt$. As I am not considering friction, the velocity in the horizontal direction will remain same. You should work out the case with the friction yourself.
