# Negative Work Explained: A Guide for Chris

• SkyrimKhajiit

#### SkyrimKhajiit

Hello,

My physics textbook describes negative work in cases where the force and displacement act in opposite directions. But I don't understand how work would be negative in the formula:

W = F • d • cos Θ

If the angle theta is 180 degrees, then the result of the cosine of 180 degrees would be -1. Let's say force is a negative value and displacement is a positive value (opposite vector directions). In this case, negative*negative*positive=positive?

Thank you,

Chris

In the formula W = F • d • cos Θ, F and d represent the magnitudes of the force and the displacement. F and d are never negative when using this formula.

In the formula W = F • d • cos Θ, F and d represent the magnitudes of the force and the displacement. F and d are never negative when using this formula.

Ah thanks for cleaning that up...

But why is that? Is it because we have the component of cosine theta?

It comes from how work is defined in physics. Work is the magnitude of the displacement, d, multiplied by the component of the force that is parallel to the displacement. This component of force is the magnitude of the force, F, multiplied by the cosine of the angle between the force vector and the displacement vector. This component of the force, F cos θ, can be positive, negative, or zero depending on the value of cos θ.

If you have studied the "dot product" of two vectors (also called the "scalar product"), then you can write the work as the dot product of the force vector and the displacement vector.

See: http://www.intmath.com/vectors/5-dot-product-vectors-2-dimensions.php

Not sure if this will help you intuit what negative work is but consider an object at rest on a frictionless surface. If a net force is applied in the horizontal direction (for simplicity) the only possibility is that the force can do positive work (the direction of changing motion will be in the direction of the force). This corresponds to an increase in the kinetic energy of the block.

Now imagine a block that is sliding at constant velocity on a frictionless surface. It takes a force in the opposite direction to slow the block down. Such a force would do negative work and decrease the kinetic energy.

It may also help to think of positive and negative work in terms of an energy transfer. If a spring brings a block to rest you can say that negative work was done by the force of the spring on the block (energy transferred from the block to the spring). Alternatively you could say that positive work was done by the block on the spring (the force exerted by the block on the spring is in the same direction as the displacement until the block changes direction).