Work done by an object on the application of a force would be zero if the displacement of the object is:
zero
Understanding Work Done and Zero Displacement
Work done by a force on an object is a fundamental concept in physics. It is defined as the product of the magnitude of the force, the magnitude of the displacement, and the cosine of the angle between the force and displacement vectors.
The formula for work done (\(W\)) is:
\[ W = \vec{F} \cdot \vec{d} = F d \cos \theta \]
Where:
- \(F\) is the magnitude of the force applied.
- \(d\) is the magnitude of the displacement of the object.
- \(\theta\) is the angle between the force vector (\(\vec{F}\)) and the displacement vector (\(\vec{d}\)).
The question asks under what condition regarding the displacement of the object, the work done by an applied force would be zero. Let's analyze the formula to find when \(W\) can be zero:
Work done \(W\) is zero if any of the following conditions are met:
- The force \(F\) is zero (\(F=0\)).
- The displacement \(d\) is zero (\(d=0\)).
- The angle \(\theta\) between the force and displacement is 90 degrees (\(\theta = 90^\circ\)), because \(\cos(90^\circ) = 0\). This means the force is perpendicular to the displacement.
The question specifically focuses on the condition related to the displacement of the object, assuming a force is applied. Let's look at the options provided in the context of the displacement \(d\).
- Positive displacement: If there is a positive displacement and a force component exists in the direction of displacement (i.e., \(\theta\) is between 0 and < 90 degrees), the work done is positive. \(W = F d \cos \theta > 0\).
- Neutral displacement: This is not a standard physics term for displacement magnitude.
- Zero displacement: If the displacement \(d\) is zero, substitute \(d=0\) into the work formula: \(W = F \times 0 \times \cos \theta\). This results in \(W = 0\), regardless of the force applied or the angle.
- Negative displacement: Displacement is a vector, so 'negative displacement' typically means displacement in the opposite direction of a chosen positive direction. If there is displacement and the force component is opposite to the displacement (i.e., \(\theta\) is between > 90 and 180 degrees), the work done is negative. \(W = F d \cos \theta < 0\) (since \(\cos \theta\) is negative for \(\theta\) between 90 and 180 degrees).
From the analysis of the formula and the options, it is clear that if the displacement of the object is zero, the work done by the applied force is zero. This is a key principle in understanding work in physics.
Consider an example: Pushing against a wall. You apply a force, but the wall does not move. Since there is no displacement (\(d=0\)), the work done by your applied force on the wall is zero.
Let's summarize the effect of displacement on work done, assuming a non-zero force is applied:
| Displacement (d) | Angle (\(\theta\)) between Force and Displacement | Work Done (W) |
|---|---|---|
| Non-zero | \(0^\circ \le \theta < 90^\circ\) | Positive (\(W > 0\)) |
| Non-zero | \(\theta = 90^\circ\) | Zero (\(W = 0\)) |
| Non-zero | \(90^\circ < \theta \le 180^\circ\) | Negative (\(W < 0\)) |
| Zero | Any angle | Zero (\(W = 0\)) |
Based on this table and the formula, the work done by an object on the application of a force would be zero if the displacement of the object is zero.
Revision Table: Key Concepts of Work Done
| Concept | Description | Formula/Condition |
|---|---|---|
| Definition of Work | Transfer of energy by force over a distance. | \(W = F d \cos \theta\) |
| Positive Work | Force has a component in the direction of displacement. | \(0^\circ \le \theta < 90^\circ\) |
| Negative Work | Force has a component opposite to the direction of displacement. | \(90^\circ < \theta \le 180^\circ\) |
| Zero Work | No displacement, force is perpendicular to displacement, or force is zero. | \(d=0\), or \(\theta=90^\circ\), or \(F=0\) |
| Units of Work | Joule (J) in SI system. | \(1 \text{ J} = 1 \text{ N} \cdot 1 \text{ m}\) |
Additional Information: Work Done in Various Scenarios
Understanding when work is done and when it is zero is crucial in physics. Here are a few more scenarios where work done is zero:
- Force is perpendicular to displacement: As mentioned, if \(\theta = 90^\circ\), \(\cos 90^\circ = 0\), so \(W=0\). Example: Work done by the centripetal force on an object moving in a circle is zero because the force is always directed towards the center (perpendicular to the instantaneous displacement which is tangential).
- No force applied: If \(F=0\), then \(W=0\), regardless of displacement. If an object is moving at a constant velocity on a frictionless surface, no net force is applied in the direction of motion, and thus, no net work is done by external horizontal forces (although work might have been done to get it moving).
- Zero displacement: This is the condition highlighted in the question. If the object does not move from its initial position (\(d=0\)), no work is done by any force acting on it. Example: Holding a heavy weight stationary. You apply a force, but since there is no displacement, the work done by your force is zero.
It's important to distinguish between applying a force and doing work. Applying a force is necessary, but not sufficient, for work to be done. There must also be displacement, and the force must have a component along the direction of displacement.
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