# Conceptually, what is negative work?

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?

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Displacement (and movement) is not always caused by the force that you want to find it's work. In your example of lowering a box, gravity must be considered. –  Mostafa May 24 '13 at 22:21
Your book may be wrong. It's not going down that makes a negative work, but, as explained by joshphysics, it's the slowing down of the movement. –  fffred May 25 '13 at 7:49
@ffred The book is right, since holding an object in a gravitational field results in a normal force from the hand, and since the motion is downward it is in fact negative work –  yo hal 2 days ago

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 the force has a component perpendicular to the direction of motion, and a component opposite 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.

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