Pure rotational motion on an incline

Question

Say a disc is kept on an incline. If the force of friction acting on it equals the component of gravitational force along the incline, the disc must not have linear acceleration, but then the force of friction exerts a torque about the body's center of mass.
If it has some angular acceleration but zero linear acceleration , the disc would just perform pure rotational motion about it's center of mass.
I just can't visualise a disc rotating about it's center of mass without moving down, where am I going wrong?

R.W. Bird · Accepted Answer

If the block is free to move, the static friction will adjust itself so that a = Rα and it will be less than the component of gravity. If some external force prevents motion, the friction force will be zero. If the disk were on a fixed axle, your condition could be met by moving the the surface of the incline up along its length.

Correct answer by R.W. Bird on April 30, 2021

Pure rotational motion on an incline

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