The work done by a constant force is represented graphically as the area under the force-displacement curve. For a constant force, this area is typically a rectangle.
Understanding Work Done by a Constant Force
Work is a fundamental concept in physics, defined as the energy transferred by a force acting over a distance. According to the reference, work is a scalar quantity and has magnitude but no direction.
When a constant force $\vec{F}$ acts on an object, causing a displacement $\vec{d}$, the work done ($W$) depends on the magnitude of the force, the magnitude of the displacement, and the angle ($\theta$) between the force and displacement vectors.
Mathematically, work done by a constant force is given by the dot product:
$W = \vec{F} \cdot \vec{d} = F d \cos \theta$
Here:
- $F$ is the magnitude of the force.
- $d$ is the magnitude of the displacement.
- $\theta$ is the angle between the force and displacement vectors.
The reference mentions a "constant angle θ from the straight line path of the particle as shown as Fig., then, The graphical representation of work done by a constant force is shown in Fig." While the specific figure isn't provided, the standard graphical representation involves plotting the component of force responsible for the work against the displacement. The reference also states "dw = F cos θ", which can be interpreted as the component of the force acting in the direction of the displacement, which is constant for a constant force and angle.
Graphical Representation: Force vs. Displacement
To represent work done by a constant force graphically, we typically use a force-displacement graph:
- The horizontal axis represents the displacement ($d$).
- The vertical axis represents the component of the force in the direction of the displacement, which is $F \cos \theta$.
Since both the force magnitude ($F$) and the angle ($\theta$) are constant, the component $F \cos \theta$ is also constant. This means the graph of $F \cos \theta$ versus $d$ is a horizontal line at the value $F \cos \theta$.
Calculating Work from the Graph
The fundamental principle of graphical representation of work states that the work done is equal to the area under the force-displacement curve.
For a constant force component $F \cos \theta$ acting over a displacement $\Delta d = d{final} - d{initial}$, the graph is a horizontal line, and the area under this line is a rectangle.
The area of this rectangle is calculated as:
$\text{Area} = \text{Height} \times \text{Width}$
$\text{Area} = (F \cos \theta) \times (\Delta d)$
Since the area under the curve represents the work done:
$W = \text{Area} = F \cos \theta \times \Delta d$
This graphical method confirms the analytical formula for work done by a constant force.
Example
Consider a constant horizontal force of 10 N pushing an object horizontally over a distance of 5 meters. Here, the force and displacement are in the same direction, so $\theta = 0^\circ$, and $\cos 0^\circ = 1$. The component of force in the direction of displacement is $F \cos \theta = 10 \text{ N} \times 1 = 10 \text{ N}$.
The force-displacement graph would be a horizontal line at 10 N on the force axis. The displacement would range from 0 m to 5 m on the displacement axis.
| Quantity | Value |
|---|---|
| Constant Force ($F$) | 10 N |
| Displacement ($\Delta d$) | 5 m |
| Angle ($\theta$) | $0^\circ$ |
| Force Component ($F \cos \theta$) | 10 N |
| Graph Shape | Rectangle |
| Area Height | 10 N |
| Area Width | 5 m |
| Calculated Work (Area) | 50 N·m (or J) |
The work done, calculated as the area of the rectangle, is $10 \text{ N} \times 5 \text{ m} = 50 \text{ J}$. This matches the formula $W = F d \cos \theta = 10 \text{ N} \times 5 \text{ m} \times \cos 0^\circ = 50 \text{ J}$.
In summary, the graphical representation of work done by a constant force involves plotting the constant component of the force parallel to the displacement against the displacement. The work done is then found by calculating the area of the resulting rectangle under the graph.
