The angle in the work calculation becomes 90° when the force applied is perpendicular to the direction of displacement. In such a case, since cos(90°) equals zero, the work done is zero. This situation occurs in scenarios like carrying a heavy bag while walking on a flat surface. The force exerted by a person is vertical (to support the weight of the bag), while the displacement is horizontal (the direction of walking). Since these directions are perpendicular, the work done by the person in holding the bag up (ignoring walking) is technically zero. This concept is crucial in distinguishing between situations where energy is being used to do work versus merely exerting a force.

In the context of physics, if there is no displacement, no work is done on an object, regardless of the amount of force applied. The fundamental formula for work, W = F * s, makes this clear, as the work done becomes zero if the displacement (s) is zero. This is true even in scenarios where a significant force is applied, such as pushing against a solid wall. Although effort is exerted, because the wall does not move, no work is technically done. This principle highlights that work in physics is as much about the effect of a force (i.e., displacement) as it is about the force itself.

The concept of work differs significantly between gravitational and frictional forces due to their different natures. When work is done against gravity, such as lifting an object, the work done is stored as gravitational potential energy. This process is generally reversible, meaning that the energy can be fully recovered, for example, when the object falls back down. In contrast, work done against frictional forces, like pushing an object across a rough surface, converts the mechanical energy into heat, which is typically dispersed into the environment and thus not recoverable. Friction, being a non-conservative force, dissipates energy, whereas gravity, a conservative force, allows for energy recovery.

In circular motion, such as swinging a pendulum, the concept of work can be more complex. When a pendulum swings, the force due to gravity acts downwards, while the displacement of the pendulum bob is along the arc of its swing. The angle between the force and displacement continuously changes throughout the motion. At the highest points of the swing, the force of gravity acts perpendicular to the displacement (angle 90°), meaning no work is done at these points. However, as the pendulum moves through its arc, gravity does work, converting potential energy into kinetic energy and vice versa. The work done in circular motion scenarios often involves calculating the components of force that are parallel to the displacement at various points, making these calculations more intricate than linear scenarios.

When the force applied is in the opposite direction to the displacement, the angle theta in the work formula W = F * s * cos(theta) becomes 180°. The cosine of 180° is -1, reflecting this opposition in direction. Therefore, the work done becomes negative, indicating that the force is acting against the direction of displacement. For instance, if a force is applied to pull an object back while it is moving forward, the work done by that force is negative. This scenario is common in situations where forces act to decelerate or stop a moving object. The negative work conceptually represents the removal or reduction of energy from the system.