Q.A round object of mass $M$ and radius $R$ rolls down without slipping along an inclined plane. The frictional force,

Step 1. Identify the role friction plays in rolling without slipping.

For a round object rolling down an incline, only static friction acts at the contact point (not kinetic friction, since there is no relative sliding between the object's surface and the incline at the point of contact). This static friction produces the torque $\tau = Rf$ that is precisely what makes the object's angular velocity increase in step with its linear velocity, maintaining the rolling condition $v=R\omega$ throughout — i.e., without this frictional torque, gravity alone would only accelerate the object translationally (as in frictionless sliding), and it would never start spinning at all.

Step 2. Check whether friction does work / dissipates energy.

Work done by a force depends on the displacement of the point of application in the direction of the force. Because there is no slipping, the contact point of the rolling object is instantaneously at rest relative to the incline (zero relative velocity) at every moment — so even though a frictional force acts there, it acts through zero relative displacement, and does zero work. No heat is generated in pure rolling (this is exactly why rolling resistance is often idealized as zero in these problems, unlike sliding friction, which does dissipate energy as heat).

Step 3. So what does friction actually accomplish, if it does no work? …

Role of static friction (zero work, nonzero torque) in maintaining v = …

• Assuming friction always dissipates energy as heat — that's true for kinetic (sliding) friction, but static friction in pure rolling does zero work since there's no relative sliding at the contact point. …