Conceptually, what is negative work?

Question

I'm having some trouble understanding the concept of negative work. For example, my book says that if I lower a box to the ground, the box does positive work on my hands and my hands do negative work on the box. So, if work occurs when a force causes displacement, how does negative work happen? Are my hands displacing anything?

Answer 1

In the context of classical mechanics as you describe, negative work is performed by a force on an object roughly whenever the motion of the object is in the opposite direction as the force. This "opposition" is what causes the negative sign in the work. Such a negative work indicates that the force is tending to slow the object down i.e. decrease its kinetic energy.

To be more mathematically precise, suppose that an object undergoes motion along a straight line (like in your example) under the influence of a force $\mathbf F$, then the work done on the object as it undergoes a small displacement $\Delta\mathbf x$ is $$ W = \mathbf F\cdot\Delta\mathbf x $$ where boldface means that the variable is a vector, and the dot represents dot product. From the definition of the dot product, we have $$ W = F\Delta x\cos\theta $$ Where $F$ is the magnitude of $\mathbf F$, $\Delta x$ is the magnitude of $\Delta \mathbf x$, and $\theta$ is the angle between $\mathbf F$ and $\Delta\mathbf x$. Note, in particular that the magnitudes are positive by definition, so the $\cos\theta$ is negative if and only of $\theta$ is between $90^\circ$ and $180^\circ$. When the angle has these ranges, the force has a component perpendicular to the direction of motion, and a component opposite to the direction of motion. The perpendicular component contributes nothing to the work, and the component opposite the motion contributes a negative amount to the work.

Answer 2

Work is the component of force parallel to the direction of motion times the displacement. That component of force could of course point in the opposite direction of motion (anti-parallel). The work done by the force is positive in the first case and negative in the second. For instance, the direction of the force of gravity on a freely falling body (dropped from rest) points towards the center of the Earth which is also the direction of displacement while falling. Hence, the force of gravity is said to be do positive work on the falling body. The falling body also experiences a upwards drag force due to air resistance. Since the the drag force is being applied in the direction opposite to that of the motion, it's said to be doing negative work on the body.