Work, Power, and Energy – Part 1: Work & Force Dynamics
📌 Table of Contents
1. Definition of Work
In physics, Work is defined as the scalar quantity that measures the energy transferred to or from an object when an external force causes a displacement of that object along the line of action of the force component.
Mathematical Definition: Work done ($W$) by a constant force is defined as the dot product (scalar product) of the force vector ($\vec{F}$) and displacement vector ($\vec{d}$):
$$W = \vec{F} \cdot \vec{d} = F d \cos\theta$$
Where:
• $F = |\vec{F}|$ is the magnitude of the applied force.
• $d = |\vec{d}|$ is the magnitude of the displacement.
• $\theta$ is the angle between the force vector $\vec{F}$ and the displacement vector $\vec{d}$.
2. Physical Characteristics of Work
| Property | Specification / Value |
|---|---|
| Governing Formula | $W = \vec{F} \cdot \vec{d} = F d \cos\theta$ |
| SI Unit | Joule (J) ($1\text{ J} = 1\text{ N}\cdot\text{m} = 1\text{ kg}\cdot\text{m}^2/\text{s}^2$) |
| CGS Unit | Erg ($1\text{ erg} = 1\text{ dyne}\cdot\text{cm} = 1\text{ g}\cdot\text{cm}^2/\text{s}^2$) |
| Unit Conversion | $$1\text{ Joule} = 10^7\text{ Ergs}$$ |
| Dimensional Formula | $[M^1 L^2 T^{-2}]$ |
| Physical Quantity Type | Scalar Quantity (It possesses magnitude only) |
3. Work Done by a Constant Force & Case Analysis
A force is classified as a constant force when both its magnitude and direction remain unchanged throughout displacement.
Figure 1: Representation of Force $\vec{F}$ acting at an angle $\theta$ relative to displacement $\vec{d}$.
Detailed Case Analysis based on Angle $\theta$:
| Case | Angle ($\theta$) | $\cos\theta$ Value | Work Done Formula | Nature & Result |
|---|---|---|---|---|
| Case 1 | $\theta = 0^\circ$ | $\cos 0^\circ = +1$ | $W = +Fd$ | Maximum Positive Work |
| Case 2 | $0^\circ < \theta < 90^\circ$ | $\cos\theta > 0$ | $W = Fd\cos\theta$ | Positive Work |
| Case 3 | $\theta = 90^\circ$ | $\cos 90^\circ = 0$ | $W = 0$ | Zero Work |
| Case 4 | $90^\circ < \theta < 180^\circ$ | $\cos\theta < 0$ | $W = -Fd|\cos\theta|$ | Negative Work |
| Case 5 | $\theta = 180^\circ$ | $\cos 180^\circ = -1$ | $W = -Fd$ | Maximum Negative Work |
4. Types of Work: Positive, Negative, and Zero Work
A. Positive Work ($W > 0$)
Work is positive when the force vector component acts in the same direction as displacement ($0^\circ \le \theta < 90^\circ$). Positive work increases energy.
- Example 1: Work done by gravity on a body falling freely toward Earth ($\theta = 0^\circ$).
- Example 2: Work done by a horse pulling a cart horizontally ($\theta = 0^\circ$).
- Example 3: Work done by a stretching force applied to expand a spring.
B. Negative Work ($W < 0$)
Work is negative when the applied force component opposes the direction of displacement ($90^\circ < \theta \le 180^\circ$). Negative work removes energy.
- Example 1: Work done by friction force on a sliding block ($\theta = 180^\circ$).
- Example 2: Work done by gravity on an object thrown vertically upward ($\theta = 180^\circ$).
- Example 3: Work done by resistive fluid drag on a sinking stone.
C. Zero Work ($W = 0$)
Work done is zero if force is zero ($F=0$), displacement is zero ($d=0$), or force is perpendicular to displacement ($\theta = 90^\circ$).
- Condition 1 ($d = 0$): Pushing against a solid wall without moving it.
- Condition 2 ($\theta = 90^\circ$): Work done by centripetal force on an orbiting satellite.
- Condition 3 ($\theta = 90^\circ$): Carrying a heavy load on one’s head while walking horizontally across a flat platform.
5. Work Done by a Variable Force
When the force acting on an object continuously changes magnitude or direction over position, the standard formula $W = F d \cos\theta$ cannot be applied directly.
A. Mathematical Expression (Calculus Approach)
Summing all small work elements from initial position $\vec{r}_i$ to final position $\vec{r}_f$ gives:
$$W = \int_{\vec{r}_i}^{\vec{r}_f} \vec{F} \cdot d\vec{r} = \int_{x_i}^{x_f} F_x \, dx + \int_{y_i}^{y_f} F_y \, dy + \int_{z_i}^{z_f} F_z \, dz$$
B. Graphical Interpretation ($F\text{-}x$ Graph Area)
The total work done by a variable force equals the area bounded by the Force-Position ($F\text{-}x$) curve and the position axis between limits $x_i$ and $x_f$.
Figure 2: Area under the $F(x)$ vs. $x$ graph representing total work done.
6. Conservative and Non-Conservative Forces
A. Conservative Force
A force is conservative if the work done by or against it in moving a particle between two points depends solely on the initial and final positions, completely independent of the path taken.
Closed Loop Property: The total work done by a conservative force along any closed path is strictly zero:
$$\oint \vec{F}_{\text{cons}} \cdot d\vec{r} = 0$$
Examples: Gravitational force, Electrostatic force, Spring force, Magnetic force.
B. Non-Conservative Force
A force is non-conservative if the work done by or against it depends directly on the path taken during displacement.
Closed Loop Property: The total work done along a closed path is non-zero ($\oint \vec{F}_{\text{non-cons}} \cdot d\vec{r} \neq 0$).
Examples: Friction force, Viscous drag force, Air resistance.
Comparison: Conservative vs. Non-Conservative Forces
| Feature | Conservative Force | Non-Conservative Force |
|---|---|---|
| Path Dependency | Independent of path taken | Dependent on path taken |
| Work in Closed Loop | Strictly Zero ($\oint \vec{F} \cdot d\vec{r} = 0$) | Non-Zero ($\oint \vec{F} \cdot d\vec{r} \neq 0$) |
| Energy Preservation | Total Mechanical Energy is conserved | Mechanical Energy is converted into heat/sound |
| Potential Energy Function | Can be defined ($F = -\frac{dU}{dx}$) | Cannot be defined |