# Can work be negative?

1. Feb 25, 2014

### thomero

Hello, i am doing an assignment and there is a research question that i need to answer.
The question is: Can work be negative?

From what i search on the internet these are my thoughts:
Work is defined by W=F.D
But work is not a vector, because it has units of energy and energy isnt vector and also it has no direction. So work can be negative in an exercise until the exercise is solved. Work is positive.

Is that right? And if not can you help me please? Thank you.

2. Feb 25, 2014

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 under the influence of a force $F$, then the work done on the object as it undergoes a small displacement $Δx$ is
$W=F.Δx$​
Dot represents dot product. From the definition of the dot product, we have
$W=F.Δxcos\theta$​

Where $F$,is the magnitude of $F$ and $Δx$ is the magnitude of $Δx$,and $\theta$ is the angle between $F$ and $Δx$

Note, in particular that the magnitudes are positive by definition, so the $cos\theta$ is negative if and only if $\theta$ is between $90°$ and $180°$.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.

Last edited: Feb 25, 2014
3. Feb 25, 2014

Thank you