Step 1: Understanding the Concept:
In physics, "work" is defined precisely as the energy transferred to or from an object via the application of force along a displacement.
A force does not always do work; for work to be performed, there must be a component of the force acting in the direction of the displacement.
If the force is applied in a way that no part of it contributes to the movement in a specific direction, the work done in that direction is zero.
Key Formula or Approach:
The work done (\( W \)) by a constant force (\( F \)) acting on an object that undergoes a displacement (\( d \)) is given by the dot product:
\[ W = \vec{F} \cdot \vec{d} = F \cdot d \cdot \cos(\theta) \]
where \( \theta \) is the angle between the force vector and the displacement vector.
Step 2: Detailed Explanation:
Let's analyze how the angle \( \theta \) affects the work done using trigonometric values:
- When \( \theta = 0^\circ \), \( \cos(0^\circ) = 1 \). Work is maximum and positive (\( W = F \cdot d \)). This happens when force is in the same direction as motion (e.g., pushing a car forward).
- When \( \theta = 180^\circ \), \( \cos(180^\circ) = -1 \). Work is maximum and negative (\( W = -F \cdot d \)). This happens when force opposes motion (e.g., friction).
- When \( \theta = 90^\circ \), \( \cos(90^\circ) = 0 \). In this case, the formula becomes:
\[ W = F \cdot d \cdot 0 = 0 \]
Therefore, when the force is perpendicular to the direction of displacement, no work is done.
Practical Examples of Zero Work:
1. A person carrying a heavy box on their head while walking horizontally: The upward force applied by the person is perpendicular to the horizontal displacement, so the work done by the person against gravity is zero.
2. Satellite in orbit: The gravitational force of the Earth acts towards the center (centripetal), which is perpendicular to the satellite's instantaneous displacement along the circular path. Thus, gravity does no work on a satellite in a perfectly circular orbit.
Step 3: Final Answer:
Work done is zero when the force and displacement are perpendicular to each other, meaning the angle between them is \( 90^\circ \).