Understanding Negative Work Done

In physics, work is performed when a force applied to an object causes it to move a certain distance. The calculation of work involves the force applied and the displacement of the object. The nature of the work done – whether it's positive, negative, or zero – depends crucially on the relative direction between the force and the displacement.

Physics Formula for Work Done

The standard formula to calculate the work done ($W$) by a constant force ($F$) over a displacement ($d$) is given by:

$ W = F \cdot d \cdot \cos(\theta) $

Here's what each component represents:

• $W$ stands for the Work Done.
• $F$ is the magnitude of the applied Force.
• $d$ is the magnitude of the Displacement of the object.
• $\theta$ is the angle measured between the direction of the Force and the direction of the Displacement.

Analyzing Conditions for Negative Work

Negative work is done when the force acts in a way that opposes the motion or displacement of the object. Based on the work formula, this occurs when the value of $\cos(\theta)$ is negative. Let's look at the options provided:

• Condition 1: Zero Displacement

  Options 1 and 2 describe situations where there is no displacement ($d=0$). If the displacement is zero, the work done is always zero, regardless of the force applied. The formula confirms this: $W = F \cdot 0 \cdot \cos(\theta) = 0$. Thus, these options do not lead to negative work.

• Condition 2: Force and Displacement in the Same Direction

  Option 3 states that the force and displacement are in the same direction. In this case, the angle $\theta$ between them is $0^\circ$. Since $\cos(0^\circ) = 1$, the work done is calculated as $W = F \cdot d \cdot 1 = Fd$. This results in positive work because the force aids the motion.

• Condition 3: Force and Displacement in Opposite Directions

  Option 4 describes the scenario where the force and displacement are in opposite directions. This means the angle $\theta$ between the force vector and the displacement vector is $180^\circ$. The cosine of $180^\circ$ is $\cos(180^\circ) = -1$. Substituting this into the work formula gives $W = F \cdot d \cdot (-1) = -Fd$. A negative result confirms that the work done is negative. This happens, for example, when friction opposes motion.

In summary, work done by a force is negative when that force acts against the direction of the object's displacement.