10.4 Work, Energy, Work-Energy Theorem, Conservative Forces & Power

Work, Energy, Work-Energy Theorem, Conservative Forces & Power

Work and energy principles form the primary alternative to Newtonian vectorial dynamics, providing scalar methods to determine final velocities, displacements, and force interactions without requiring time-dependent acceleration integrals. In the Intermediate Physics Part I syllabus and the UET ECAT entry test, mechanical energy problems emphasize work by constant and variable forces, the work-energy theorem, conservative vector fields, gravitational and elastic potential energies, instantaneous power, and mechanical efficiency.

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1. Mechanical Work Done by Constant and Variable Forces

Work Done by a Constant Force

The mechanical work $W$ performed by a constant force $\mathbf{F}$ acting on a body that undergoes displacement $\mathbf{d}$ is defined as the scalar (dot) product of force and displacement:

$W = \mathbf{F} \cdot \mathbf{d} = F d \cos\theta$

where $\theta$ is the angle between the force vector and the displacement vector.

• Positive Work ($0^\circ \le \theta < 90^\circ$): The force possesses a component along the direction of motion, adding mechanical energy to the system (e.g., an engine pulling a carriage, $\theta = 0^\circ \implies W = Fd$).
• Zero Work ($\theta = 90^\circ$): The force acts perpendicular to displacement, contributing no work: $W = F d \cos(90^\circ) = 0$ Examples of zero mechanical work include:
  • Centripetal force acting on a satellite in circular orbit ($\mathbf{F}_c \perp \mathbf{v}$).
  • The tension in a simple pendulum string ($\mathbf{T} \perp \mathbf{v}$).
  • Normal reaction force on a crate sliding across a flat horizontal floor.
  • A porter carrying a load on his head while walking horizontally at constant speed ($mg \perp d$).
• Negative Work ($90^\circ < \theta \le 180^\circ$): The force opposes the direction of motion, removing mechanical energy from the system (e.g., kinetic friction acting at $\theta = 180^\circ \implies W = -f_k d$).

Work Done by a Variable Force

When the applied force changes in magnitude or direction along the displacement path $x$, the total work is obtained by integrating over infinitesimal displacements $dx$:

$W = \int_{x_i}^{x_f} F(x) \cos\theta \, dx$

For one-dimensional motion along the $x$-axis:

$W = \int_{x_i}^{x_f} F_x(x) \, dx$

Graphical Work Evaluation ($F-x$ Graph)

On a Force versus Position ($F-x$) graph, the definite integral equals the signed geometric area under the curve:

$W = \text{Area under the } F-x \text{ curve between } x_i \text{ and } x_f$

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2. Conservative vs. Non-Conservative Forces

Forces in nature are fundamentally divided into conservative and non-conservative forces based on their path dependence.

Conservative Forces

A force is conservative if it satisfies any of the following mutually equivalent physical definitions:

1. The work done by the force in moving a particle between any two points depends solely on the initial and final endpoints and is completely independent of the path taken: $W_{A \to B}^{(\text{Path 1})} = W_{A \to B}^{(\text{Path 2})} = W_{A \to B}^{(\text{Path 3})}$
2. The work done by the force around any closed loop is identically zero: $\oint \mathbf{F} \cdot d\mathbf{r} = 0$
3. The force can be expressed as the negative spatial gradient of a scalar potential energy function $U$: $\mathbf{F} = -\nabla U \implies F_x = -\frac{\partial U}{\partial x}$

Examples of Conservative Forces:

• Gravitational force: $\mathbf{F}_g = -\frac{G M m}{r^2}\hat{\mathbf{r}}$
• Electrostatic Coulomb force: $\mathbf{F}_e = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}\hat{\mathbf{r}}$
• Elastic restoring force of an ideal Hookean spring: $F_s = -kx$

Non-Conservative (Dissipative) Forces

A force is non-conservative if the work it performs depends on the actual path traversed. Around a closed path, non-conservative forces do non-zero work, dissipating mechanical energy into heat, sound, or internal deformation:

$\oint \mathbf{F}_{\text{nc}} \cdot d\mathbf{r} \ne 0$

Examples of Non-Conservative Forces:

• Friction ($f_k$): always opposes motion, doing negative work along every path segment.
• Viscous drag in fluids ($F_d = 6\pi\eta r v$).
• Air resistance.
• Tension in a flexible cord, normal contact forces with deformation, and motor propulsion forces.

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3. Kinetic and Potential Energy Formulations

Kinetic Energy ($K$)

Kinetic energy is the capacity of a body to do work by virtue of its motion:

$K = \frac{1}{2}mv^2$

Relating kinetic energy to linear momentum $p = mv$:

$K = \frac{p^2}{2m} \iff p = \sqrt{2mK}$

ECAT Percentage Change Rule for Momentum and Energy

• If momentum increases by a factor of $(1 + x)$, kinetic energy increases by a factor of $(1 + x)^2$: $\Delta K\% = [(1 + x)^2 - 1] \times 100\%$ For example, if momentum increases by $50\%$ ($x = 0.5$): $\Delta K\% = [(1.5)^2 - 1] \times 100\% = [2.25 - 1] \times 100\% = 125\%$
• If kinetic energy increases by $300\%$, the new energy is $4K$, meaning momentum increases by $\sqrt{4} - 1 = 1 = 100\%$.

Gravitational Potential Energy ($U_g$)

Gravitational potential energy is the energy stored in a body by virtue of its position in a gravitational field. Near Earth's surface (where $g$ is uniform):

$U_g = mgh$

where $h$ is measured relative to an arbitrary reference datum where $U_g = 0$.

Elastic Potential Energy of a Hookean Spring ($U_s$)

According to Hooke's Law, the restoring force exerted by a spring stretched or compressed by distance $x$ from its equilibrium position is:

$F_s = -kx$

where $k$ is the spring stiffness constant ($\text{N/m}$). The external force required to stretch the spring without acceleration is $F_{\text{ext}} = +kx$. The work performed in extending the spring from $x = 0$ to $x$ is stored as elastic potential energy:

$U_s = \int_0^x F_{\text{ext}} dx = \int_0^x kx dx = \frac{1}{2}kx^2$

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4. The Work-Energy Theorem

The Work-Energy Theorem states that the net work done by all forces (conservative, non-conservative, internal, and external) acting on a particle equals the resulting change in the particle's kinetic energy:

$W_{\text{net}} = \Delta K = K_f - K_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

Derivation from Newton's Second Law

Using $F_{\text{net}} = ma = m\left(v\frac{dv}{dx}\right)$:

$W_{\text{net}} = \int_{x_i}^{x_f} F_{\text{net}} dx = \int_{x_i}^{x_f} m\left(v\frac{dv}{dx}\right) dx = \int_{v_i}^{v_f} mv dv = \left[\frac{1}{2}mv^2\right]_{v_i}^{v_f} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

Generalized Work-Energy Principle

Separating net work into conservative work $W_c$ and non-conservative work $W_{\text{nc}}$:

$W_{\text{net}} = W_c + W_{\text{nc}} = \Delta K$

Because work done by conservative forces equals the negative change in potential energy ($W_c = -\Delta U$):

$-\Delta U + W_{\text{nc}} = \Delta K \implies W_{\text{nc}} = \Delta K + \Delta U = \Delta(K + U) = \Delta E_{\text{mech}}$

The work done by all non-conservative forces equals the change in the total mechanical energy of the system.

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5. Law of Conservation of Mechanical Energy

If only conservative forces perform work on an isolated system ($W_{\text{nc}} = 0$):

$\Delta E_{\text{mech}} = 0 \implies K_i + U_i = K_f + U_f = \text{constant}$

Total mechanical energy remains constant throughout the motion.

Energy Interconversions for a Freely Falling Body

Consider a body of mass $m$ dropped from rest at height $h$:

1. At top release point ($y = h$): $v = 0 \implies K_1 = 0, \quad U_1 = mgh \implies E_1 = mgh$
2. At intermediate elevation $y$ ($0 < y < h$): After falling a distance $(h - y)$, velocity is $v = \sqrt{2g(h - y)}$: $K_2 = \frac{1}{2}mv^2 = \frac{1}{2}m[2g(h - y)] = mg(h - y)$ $U_2 = mgy$ $E_2 = K_2 + U_2 = mg(h - y) + mgy = mgh$
3. Just before ground contact ($y = 0$): Velocity is $v = \sqrt{2gh}$: $K_3 = \frac{1}{2}m(2gh) = mgh, \quad U_3 = 0 \implies E_3 = mgh$

At every point in the descent, the sum of kinetic and gravitational potential energy equals the initial potential energy $mgh$.

Falling in the Presence of Air Friction ($f$)

When a non-conservative retarding drag force $f$ opposes motion over vertical height $h$:

$\text{Loss in Potential Energy} = \text{Gain in Kinetic Energy} + \text{Work against Friction}$

$mgh = \frac{1}{2}mv^2 + fh \implies v = \sqrt{2gh - \frac{2fh}{m}}$

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6. Power and Mechanical Efficiency

Power ($P$)

Power is the rate at which work is performed or energy is transferred.

$\text{Average Power} = P_{\text{avg}} = \frac{\Delta W}{\Delta t}$

$\text{Instantaneous Power} = P = \lim_{\Delta t \to 0}\frac{\Delta W}{\Delta t} = \frac{dW}{dt} = \frac{\mathbf{F} \cdot d\mathbf{r}}{dt} = \mathbf{F} \cdot \mathbf{v} = F v \cos\theta$

If a body moves with instantaneous velocity $\mathbf{v}$ under applied force $\mathbf{F}$, instantaneous power is the dot product of force and velocity.

Units of Work, Energy, and Power

ECAT Unit Distinction: The kilowatt-hour ($\text{kWh}$) has units of power multiplied by time; therefore, it is a unit of energy, not power.

Mechanical Efficiency ($\eta$)

Efficiency is the ratio of useful work output to total energy input, expressed as a percentage:

$\eta = \frac{\text{Useful Work Output}}{\text{Total Energy Input}} \times 100\% = \frac{\text{Power Output}}{\text{Power Input}} \times 100\%$

Because non-conservative forces like friction and viscous drag dissipate mechanical energy as heat, the efficiency of any real machine is strictly less than $100\%$ ($\eta < 1$).

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Work, Energy, and Power Relationships Reference Table

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7. Non-Conventional Energy Sources

The work–energy chapter of Intermediate physics also surveys energy sources other than fossil fuels. Each converts one form of energy into another, so the same conservation reasoning applies:

Worked example (wind): air density $\rho = 1.2\text{ kg/m}^3$, swept area $A = 100\text{ m}^2$, wind speed $v = 10\text{ m/s}$: $P = \frac{1}{2}(1.2)(100)(10^3) = 60\text{ kW}$ of kinetic energy flux (a real turbine extracts only part of this).

Worked example (hydro): $Q = 10\text{ m}^3/\text{s}$, $h = 50\text{ m}$, $\eta = 0.8$: $P = 0.8 \times 1000 \times 9.8 \times 10 \times 50 \approx 3.9\text{ MW}$.