I am able to understand the increase in translational KE. However, the increase in rotational KE doesn’t seem right to me. The spring applies a force on the center of the cylinder and therefore has zero torque about the rotational axis (which passes through the center of mass). Why does the rotational kinetic energy of the cylinder increase?
I think the easiest way to look at this problem is to look at it as realistically as possible.
- When the spring is stretched it applies a force on the center of mass of the cylinder.
- If there was no friction, the cylinder would just slide, so you would get an increase in kinetic energy equal to the amount of work done by the spring over a distance. So: $$KE_{linear} = W = k\Delta x$$
- But, since the cylinder is on a surface with friction, the cylinder will roll as well as translate. And, like @Aaron Stephens said, energy is energy, so the sum of the rotational kinetic energy plus the translational kinetic energy is still going to equal the work done with each increment of displacement. Therefore: $$KE_{linear}+KE_{angular} = k\Delta x$$
I think the confusion comes when trying to imagine that friction supplies a force that gives the cylinder rotational kinetic energy. Common sense dictates that the spring is the source of the force that causes the cylinder to roll and translate. It is not friction that supplies the force that causes the cylinder to roll.
This is obviously true (that the spring is the source of the force) since the spring is the entity in the problem that has potential energy stored within it. Static Friction doesn't supply energy to move anything. Rather, static friction is more like a bond that connects things together so they don't move.
(Note, if there is friction, and the two surfaces move in relation to each other, it's sliding friction, and that's a different/more complex problem where we have to compute the amount of heat lost too.)
Thus, the cylinder and the table are bonded together by static friction, so they can't slide over each other. So, when the spring pulls on the center of the cylinder, the table and cylinder don't slide. At that point (when the cylinder moves a tiny bit), the cylinder starts to translate and pick up some kinetic energy, but because the cylinder and table are bonded, the cylinder rolls incrementally also and picks up a little rotational kinetic energy too.
Therefore, the total amount of energy supplied by the spring contracting is divided into two compartments, 1) some rolling/angular kinetic energy, and 2) some linear/translational kinetic energy.
Rolling problems are made confusing by pretending (framing the problem in terms such) that friction supplies a motive force. Just look at the problem realistically. Identify the source of the force (which always comes from the conversion of one type of energy into another). Locate the lever arms and hinge points (the points around which motion rotates). The problem seems more intuitive when you model the problem to reflect reality.