Restoring Force

The word describes what a force does, not what produces it. AP Physics 1 essential knowledge 7.1.A.2.i puts it in one line: a restoring force is a force that is exerted in a direction opposite to the object's displacement from an equilibrium position. 7.1.A.2.ii pins down the reference point, an equilibrium position being a location at which the net force exerted on an object or system is zero.

So the test is one question about direction. Displace the object; does the net force now point back the way it came?

Restoring is not enough for simple harmonic motion. 7.1.A.2 adds a second condition: SHM results when the magnitude of the restoring force is proportional to the object's displacement from equilibrium. A force that pushes back but does not grow in proportion to the displacement fails that second test, whatever else it does. The CED gives the proportional case as a derived equation, $ma_x = -k\Delta x$, which is not itself printed on the AP Physics 1 equation sheet.

It need not be a force. 7.1.A.2.iii treats the pendulum through a restoring torque instead: the motion of a pendulum at small angular displacement can be modeled as simple harmonic motion because the restoring torque is proportional to the angular displacement. Nothing about that argument mentions a spring.

One consequence worth carrying. Because the force is proportional to displacement and points the other way, it is zero at equilibrium, where the speed is greatest, and largest at the turning points, where the speed is zero. Acceleration and displacement are always opposite in sign. Simple harmonic motion works the algebra, and Hooke's law is the spring version of the same condition.