A solid sphere (or any round object like a cylinder, disc for that matter) is kept on a frictionless surface. A force is applied to the sphere, parallel to the surface, and passing through its center of mass. Weight, Normal force and this applied force all act at CoM and hence would not produce any torque. However, the applied force would produce a torque about the Point of Contact. Normal force and Weight act through the PoC and won't produce torques to counter this one. How come then there is no rotation about PoC and instead, there shall be just translation?
Answer
If you apply an unbalanced force to the sphere, then it (and the point of contact) will accelerate. This means that a frame where the bottom of the sphere (the point of contact) is at rest is not an inertial frame.
In this accelerating frame, fictitious forces appear. In particular, $F_{fict} = -ma_{frame}$ can be assumed to act through the center of mass. This force exactly counters the applied force and you get no net torque (or net acceleration) in the frame.
If the applied force is at a different distance from the bottom of the sphere, then the torque about that point will have a different magnitude from that of the fictitious force and they won't cancel out. The rotation about that point will accelerate.
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