Work Done: Force and Displacement Relationship

In physics, the concept of work done by a force is crucial. Work is done on an object when a force acting on it causes it to move a certain distance (displacement). The amount of work done depends on the magnitude of the force, the magnitude of the displacement, and the angle between the force vector and the displacement vector.

Calculating Work Done

The work done ($W$) by a constant force ($\vec{F}$) causing a displacement ($\vec{d}$) is given by the formula:

$ W = \vec{F} \cdot \vec{d} $

This dot product can also be expressed in terms of the magnitudes of the force ($F$), the displacement ($d$), and the angle ($\theta$) between the force and displacement vectors:

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

Here:

• $F$ is the magnitude of the force.
• $d$ is the magnitude of the displacement.
• $\theta$ is the angle between the direction of the force and the direction of the displacement.

Analyzing Conditions for Zero Work Done

The work done ($W$) will be zero if any of the following conditions are met:

• The force ($F$) is zero.
• The displacement ($d$) is zero.
• The angle $\theta$ is such that $\cos(\theta) = 0$.

We are interested in the case where a non-zero force acts on an object, causing a non-zero displacement. Therefore, the condition for zero work done relies on the angle $\theta$. The value of $\cos(\theta)$ is zero when $\theta = 90^\circ$ (or $\frac{\pi}{2}$ radians).

This means that the work done by the force is zero if the displacement of the object is perpendicular to the direction of the force.

Evaluating the Options

Let's examine the given options based on the formula $W = F d \cos(\theta)$:

• Option 1: Displacement is in the opposite direction of the force.
  In this case, $\theta = 180^\circ$. Since $\cos(180^\circ) = -1$, the work done is $W = -Fd$. This is negative work, not zero.
• Option 2: Displacement is in the same direction of the force.
  Here, $\theta = 0^\circ$. Since $\cos(0^\circ) = 1$, the work done is $W = Fd$. This is positive work, not zero.
• Option 3: Displacement is in a perpendicular direction of the force.
  In this scenario, $\theta = 90^\circ$. Since $\cos(90^\circ) = 0$, the work done is $W = Fd \times 0 = 0$. This results in zero work done.
• Option 4: None of the above.
  This is incorrect because Option 3 correctly describes a condition for zero work done.

Conclusion

The work done by a force on an object is zero specifically when the displacement occurs in a direction perpendicular to the applied force. This is a fundamental concept in understanding work in physics.