How Force Direction Determines Positive, Negative, and Zero Work

Work depends on force direction

Work measures energy transfer associated with a force acting through a displacement. For a constant force,

$$W=\mathbf F\cdot\Delta\mathbf r=F\Delta r\cos\theta.$$

The cosine is what makes direction matter.

Positive work

If a force has a component in the direction of motion, $\theta<90^\circ$ and $W>0$. That force transfers energy into the object’s mechanical motion.

For example, a 40 N pull at $60^\circ$ to a 6.0 m horizontal displacement does

$$W=(40)(6.0)\cos60^\circ=120\,\text{J}.$$

Negative work

If a force points partly opposite the displacement, $\theta>90^\circ$, its work is negative. Kinetic friction on a sliding object is a common example. A 10 N friction force opposite a 6.0 m displacement does

$$W=-(10)(6.0)=-60\,\text{J}.$$

Zero work

A force does zero work when it is perpendicular to displacement because $\cos90^\circ=0$. For an object moving horizontally on a level surface, the ideal normal force is vertical, so its work is zero.

A force can also do zero work when displacement itself is zero.

Use the component parallel to motion

The equation can be read as

$$W=F_{\parallel}\Delta r.$$

This is often the fastest interpretation: resolve the force into components and keep only the component parallel to the displacement. The sign of that component immediately tells whether the work is positive, negative, or zero.