Question

The conservation of energy for rolling object is: ΔK for translation +ΔK for rotation +ΔU for gravity, why it is not 2ΔU ??

Because we add rotation and translation laws

Answer 1

Actually for any rolling body undergoing translational + rotational motion

Total energy is = K_rot+ K_trans +U_rot + U_trans

=

here they have mentioned for gravity where F = mg. The torque associated with a given force is the product of the magnitude of that force and the perpendicular distance between the line of action of the force and the axis of rotation. The weight of an rolling object acts at its center of mass. In this case, the axis of rotation passes through the center of mass. Hence, distance between the line of action of the force associated with the weight mg and the axis of rotation is zero. It follows that the associated torque is also zero. So, in these cases (like rolling on a inclined plane) when force is gravitational, net torque will be zero. So no change in potential energy due to rotational motion.