newtonian mechanics - What happens when a force is not applied on the centre of the mass?

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force applied not on the center of mass

states that for a force applied not directly on the center of mass will have the same linear acceleration as one applied on the center of mass. If this is true, then consider the following:

A disc with center of mass on the dot. Firstly, does that mean that force a will result in purely downward motion, as if a could be translated towards the center of mass (of course with some torque as well), or will it be at an angle (if so, what angle?). Secondly, does force a balance with force c (no net linear force) or does force a balance with force d (for linear) or do both pairs balance? I'm supposing that a and c will generate no torque and a and d will, but I am only really interested in linear motion for now.

Answer

Force a alone will produce both downward motion (as if the force were applied at the center of mass) and rotational motion, because it causes torque. But the linear acceleration is the same, regardeless of where you apply the force. For traslation, both c and d balance a, so the disk will not translate in either case, but it will rotate with d because there will be net torque.