Final answer:
Only scenario number 2, where a force is exerted while pulling a wagon up a hill, involves work being done because the force causes a displacement in the direction of the force. In all other scenarios, there is no displacement in the direction of the force, thus no work is performed.
Step-by-step explanation:
Work is defined in physics as the process by which energy is transferred from one system to another through the action of forces. For work to be done, there must be a displacement of the object by the force in the direction of the force. In the scenarios presented, number 2) where a force is exerted while pulling a wagon up a hill represents a situation where work is being done because there is a displacement in the direction of the force, and this action is done against gravitational force.
In contrast, no work is done in scenarios 1), 3), and 4) because there is either no displacement (as in the case of a person standing still on the surface of Earth or a tug-of-war with no movement) or the force is perpendicular to any possible displacement (as with a force acting vertically on a cart that can only move horizontally).
Thus, in physics, work involves a transformation of energy and is associated with forces that cause an object to move. This work can result in an increase in mechanical energy or be stored as potential energy, for instance when non-zero displacement occurs against gravity.