A student investigates how changing either the distance from the pivot or the angle between the force and the lever arm affects the torque exerted on a shovel whose handle is modeled as a uniform rigid rod. The student makes the following four claims while holding the magnitude of the applied force constant.
I. Doubling the distance from the pivot always doubles the torque.
II. Reducing the angle between the force and the handle from 90° to 45° cuts the torque in half.
III. Swinging the shovel so the force is applied at 60° instead of 30° to the handle increases the torque by a factor of two.
IV. Moving the applied force closer to the pivot can never increase the torque.
(a) For each claim (I–IV) state whether the claim is True or False.
(b) For every claim you identified as True, provide a clear explanation referencing the relevant geometric quantities in the torque relationship.
(c) For every claim you identified as False, describe a specific counter-example (numerical or qualitative) that contradicts the claim and explain why the contradiction occurs.