Short Notes: Work, Energy and Power

Table of Contents
1. Work and Energy Formulae
2. Definitions and Basic Concepts
3. Conservation of Mechanical Energy
4. Collisions and Momentum
5. Coefficient of Restitution
6. Work-Energy Theorem: Derivation and Remarks
7. Sign Conventions, Units and Remarks
8. Examples and Typical Applications
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This document summarises concise, syllabus-aligned notes on work, energy and power, with definitions, standard formulae, theorem derivations and typical applications. Key symbols used throughout: m - mass; v, u - final and initial speeds; F - force; s, x, r - displacement or position; g - acceleration due to gravity; k - spring constant; t - time; θ - angle between force and displacement vectors; e - coefficient of restitution.

Work and Energy Formulae

Definitions and Basic Concepts

• Work done by a force on a body during a displacement is the scalar product of force and displacement. It is positive when the force has a component along the displacement and negative when it opposes displacement.
• Energy is the capacity to do work. Mechanical energy commonly appears as kinetic energy (energy of motion) and potential energy (stored energy due to position or configuration).
• Power is the rate at which work is done or energy is transferred; instantaneous power is the dot product of force and velocity.
• Conservative force is a force for which work done between two points is independent of the path taken (examples: gravity, ideal spring force). For conservative forces there exists a potential energy function.
• Non-conservative force is a force for which work depends on the path (examples: friction, air resistance); such forces dissipate mechanical energy, usually into heat.

Conservation of Mechanical Energy

Principle

For a system acted on only by conservative forces, the total mechanical energy remains constant:

$E=\mathrm{KE}+\mathrm{PE}=\text{constant}$

Standard applications

• Free fall from rest: If a mass falls from height h with initial speed zero, potential converts to kinetic: $mgh=\tfrac{1}{2}mv^{2}$, giving $v=\sqrt{2gh}$.
• Vertical motion with initial speed: For a mass projected vertically, use $mgh+\tfrac{1}{2}mu^{2}=\tfrac{1}{2}mv^{2}$ with sign conventions for height and velocity taken consistently.
• Spring-mass system (simple harmonic motion extremes): Maximum potential energy in the spring at amplitude A equals maximum kinetic energy at equilibrium: $\tfrac{1}{2}kA^{2}=\tfrac{1}{2}mv_{\max}^{2}$. At displacement x: $\tfrac{1}{2}kA^{2}=\tfrac{1}{2}kx^{2}+\tfrac{1}{2}mv^{2}$.
• Non-conservative work: If non-conservative forces do work $W_{\text{nc}}$, the energy balance is $E_{\text{final}}-E_{\text{initial}}=W_{\text{nc}}$. For friction, $W_{\text{nc}}$ is negative and mechanical energy decreases.

Collisions and Momentum

Collisions are rapid interactions between bodies. For an isolated system, linear momentum is always conserved, while kinetic energy may or may not be conserved depending on collision type.

Conservation of linear momentum for two bodies of masses $m_{1}$ and $m_{2}$ with initial velocities $u_{1},u_{2}$ and final velocities $v_{1},v_{2}$ gives:

$m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}$

Combine this with energy conservation (when applicable) or restitution relations to solve collision problems.

Coefficient of Restitution

• Definition: $e=\dfrac{\text{velocity of separation}}{\text{velocity of approach}}$.
• One-dimensional two-body form: $e=\dfrac{v_{2}-v_{1}}{u_{1}-u_{2}}$, where indices 1 and 2 label the bodies and velocities are along the line of impact, with consistent signs.
• Values and meaning: $e=1$ corresponds to a perfectly elastic collision; $0<><> to a partially inelastic collision; $e=0$ to a perfectly inelastic collision (bodies stick together).
• Useful property for elastic collisions: Relative speed of approach equals relative speed of separation when $e=1$.

Work-Energy Theorem: Derivation and Remarks

The work-energy theorem relates the net work done on a particle to the change in its kinetic energy. The following derivation assumes motion along a straight line under a net force $\vec F$.

$\vec F = m\vec a$

$\vec a = \dfrac{d\vec v}{dt}$

Dot both sides with $\vec v$:

$\vec F\cdot\vec v = m \dfrac{d\vec v}{dt}\cdot\vec v$

Recognise the right-hand side as a time derivative:

$m \dfrac{d\vec v}{dt}\cdot\vec v = \dfrac{d}{dt}\!\left(\tfrac{1}{2} m v^{2}\right)$

Integrate both sides from initial time $t_{1}$ to final time $t_{2}$:

$\displaystyle\int_{t_{1}}^{t_{2}} \vec F\cdot\vec v\,dt = \int_{t_{1}}^{t_{2}} \dfrac{d}{dt}\!\left(\tfrac{1}{2} m v^{2}\right) dt$

Change the left integral variable using $d\vec r = \vec v\,dt$ to obtain the work integral:

$\displaystyle\int_{r_{1}}^{r_{2}} \vec F\cdot d\vec r = \tfrac{1}{2} m v^{2}\Big|_{t_{1}}^{t_{2}}$

Thus the net work done equals the change in kinetic energy:

$W=\Delta\mathrm{KE}=\tfrac{1}{2}m(v^{2}-u^{2})$

Sign Conventions, Units and Remarks

• Units: Unit of work and energy is the joule (J), where 1 J = 1 N·m. Unit of power is the watt (W), where 1 W = 1 J s⁻¹.
• Sign convention for work: Work done by a force that increases the kinetic energy of the system is positive; work done by a force that reduces kinetic energy is negative.
• Path dependence: For conservative forces, work depends only on end points; for non-conservative forces, work depends on the path.
• Work done by gravity: Gravity is conservative and path independent. For a vertical displacement from height y_{1} to y_{2} (with upward chosen positive), the work by gravity is $W_{\text{gravity}}=-mg(y_{2}-y_{1})$, and a change in gravitational potential energy is $\Delta \mathrm{PE}=mg(y_{2}-y_{1})$.
• Work done by a spring force: For an ideal linear spring, work done by the spring when moving from displacement $x_{1}$ to $x_{2}$ is $W_{\text{spring}}=-\dfrac{1}{2}k(x_{2}^{2}-x_{1}^{2})$, and the potential energy stored is $\tfrac{1}{2}kx^{2}$.
• Instantaneous power useful form: $P=\vec F\cdot\vec v$ so when a constant vertical force $mg$ lifts a mass at constant speed $v$, the required power is $P=mgv$.

Examples and Typical Applications

• Free fall from rest: A block dropped from height h reaches speed v at the bottom with $v=\sqrt{2gh}$, obtained from $mgh=\tfrac{1}{2}mv^{2}$.
• Compression of a spring: A mass compresses a spring of constant k by amount x; the work done on the spring equals the energy stored, $\tfrac{1}{2}kx^{2}$. If released, this energy converts into kinetic energy of the mass (neglecting losses).
• Power required to lift a mass: To lift a mass m at constant speed v vertically, required lifting force equals mg and instantaneous power is $P=mgv$.
• Collision problems: Use conservation of momentum and, if elastic, conservation of kinetic energy or restitution relations to find final velocities. For perfectly inelastic collisions use momentum conservation with combined mass after impact.

If worked examples with stepwise solutions are required for particular problem types (free fall, spring-mass systems, collisions, power calculations), these can be supplied with clear stepwise mathematics and explanations on request.