I'm studying the rolling motion, but, Why the torque by static friction does work? If the point of application is at rest relative to the inclined plane, therefore, the point of application doesn't move.
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The work done by friction depends on whether the body is rolling without slipping, or slipping. For rolling without slipping the net work done by friction is zero.
For your problem, the object rolls without slipping and only gravity does work. The work done by friction consists of two parts: work for translational motion of the center of mass (negative) and work for rotation motion about the center of mass (positive). The net work done by friction is the sum of these two terms and is zero for pure rolling with no slipping. Your problem shows the work done by gravity and friction for translation of the center of mass as: $mgh-F_rx$; $mgh$ is the work by gravity and $-F_rx$ is the work by friction. Your problem shows the work done by friction for rotation about the center of mass as: $F_r R\phi = F_rx$; this work is due to the torque from the force of friction (gravity has no torque about the center of mass). The total (net) work is the sum of the work for translation plus the work for rotation and is $W = (mgh-F_rx) + (F_rx) =mgh$ as your problem states; note that the net work by friction is zero because the two terms for the work by friction involving translation and rotation sum to zero.
For slipping, friction does net work.
See Consistent Approach for Calculating Work By Friction for Rigid Body in Planar Motion for a detailed discussion of the work done by friction for pure rolling and for slipping.
John Darby
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Without friction the wheel would not roll down but rather slide down. Friction acts to ‘catch’ the wheel in the tangent direction shown in your diagram.
From the wheel’s perspective, something is making it rotate which means Work is being done on the wheel.
