Positive, Negative, and Zero Work

Work done by a force is positive when the force and displacement act in the same direction. For example, when a person pushes a box up a slanting plane, the applied force and displacement are aligned, resulting in positive work. The work is maximum when the angle between the force and displacement vectors is zero degrees.

The work done depends on the angle between force and displacement vectors. When the angle is 0°, work is maximum and positive. At 90°, work is zero because the force is perpendicular to displacement. Between 90° and 180°, work is negative as the force opposes displacement. The magnitude of force multiplied by the cosine of this angle determines the force component in the displacement direction.

Negative work occurs when force acts opposite to displacement. When a block slides back down a slanting plane, the applied force opposes the direction of motion, producing negative work. Similarly, friction acting against an object's movement does negative work, removing energy from the system.

Work is zero when the force is perpendicular to the displacement, meaning the angle between them is 90 degrees. For instance, when a person carries a box horizontally, the upward force they apply is perpendicular to the horizontal displacement, so no work is done by that force despite the object moving.

Work done by a constant force equals the integral of force with respect to displacement along the path. The work depends on the force component in the direction of displacement and the magnitude of that displacement. Forces can vary as a function of position, requiring integration along the actual path taken between two points.

Work classification depends on the relationship between force and displacement vectors. Work is positive when force is generally in the direction of displacement, negative when opposite, and zero when perpendicular. The force component in the displacement direction, determined by the cosine of the angle between them, directly determines the work's sign and magnitude.

When force and displacement are perpendicular, the angle between them is 90 degrees. The cosine of 90° equals zero, making the force component in the displacement direction zero. Since work depends on this component multiplied by displacement, the result is zero work regardless of the force or displacement magnitudes.