Topic 4
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
Topic 4 — Key Facts for DU Admission (Bangladesh) Core concept: Work, Energy, and Power — forms of energy, energy transformations, and the rate at which energy is transferred or work is done High-yield point: Work-energy theorem, conservation of mechanical energy, kinetic and potential energy calculations ⚡ Exam tip: Numerical problems combining work-energy theorem with conservation of energy appear in nearly every DU admission physics paper; practice the formula W = Fd cosθ thoroughly
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Topic 4 — DU Admission (Bangladesh) Study Guide Overview: Work, energy, and power form the conceptual bridge between force and motion, introducing energy as a scalar quantity that simplifies problem solving Core principles: Work-energy theorem, conservation of mechanical energy, different forms of energy and their interconversion Key points: Kinetic energy and gravitational potential energy formulas, spring potential energy, power definitions and efficiency calculations Study strategy: Focus on energy transformation diagrams for each physical situation; practice connecting force-displacement analysis with energy analysis
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer study timeline.
Work, Energy, and Power — Complete Study Notes
The Concept of Work
In physics, work possesses a precise technical meaning quite different from its everyday usage. Work is done when a force applied to a body produces motion in the direction of that force. If the force and displacement are not collinear, only the component of the force along the displacement direction contributes to the work done.
The fundamental work formula:
W = F · d = Fd cosθ
Where F is the magnitude of the applied force, d is the displacement of the point of application, and θ is the angle between the force vector and the displacement vector.
The geometric interpretation is straightforward: W equals the product of the force magnitude, the displacement magnitude, and the cosine of the angle between them. This equals the scalar (dot) product of force and displacement vectors.
Special angle cases:
| Angle (θ) | cosθ | Work | Physical Situation |
|---|---|---|---|
| 0° | 1 | Maximum (+Fd) | Force and displacement perfectly aligned |
| 90° | 0 | Zero | Force perpendicular to displacement |
| 180° | −1 | Minimum (−Fd) | Force opposes motion |
| 60° | 0.5 | +½Fd | Force at acute angle to displacement |
| 120° | −0.5 | −½Fd | Force at obtuse angle opposing motion |
Work is a scalar quantity — it has magnitude but no direction. This is a significant advantage over force analysis because scalar quantities add without regard to direction, simplifying multi-force problems.
Sign conventions for work:
- Positive work (+): The force component acts in the direction of motion, increasing the body’s energy
- Negative work (−): The force component opposes motion, extracting energy from the body (examples: friction, air resistance, gravity when moving upward)
- Zero work (0): The force is perpendicular to motion — the force does not affect the body’s kinetic energy in this direction (example: centripetal force in uniform circular motion)
SI unit: Joule (J)
- 1 Joule = work done when a force of 1 Newton moves an object 1 meter in the direction of the force
- 1 J = 1 N × 1 m = 1 kg·m²/s²
- CGS unit: Erg = 10⁻⁷ J
- Nutrition unit: Calorie = 4.184 J
- Atomic physics unit: Electron-volt (eV) = 1.6 × 10⁻¹⁹ J
Forms of Energy
Energy is the capacity to do work. Like work, energy is a scalar quantity. Energy exists in many forms; the DU admission syllabus focuses primarily on mechanical energy (kinetic and potential).
Kinetic Energy (KE)
The energy a body possesses by virtue of its motion.
Formula: KE = ½mv²
Where m is the body’s mass and v is its velocity. Since velocity squared is always non-negative, kinetic energy is always positive or zero — it can never be negative. Doubling the mass doubles the KE; doubling the velocity quadruples the KE (because KE scales with v²).
Physical interpretation: The work required to accelerate a body from rest to velocity v equals ½mv². Conversely, bringing a moving body to rest requires negative work equal in magnitude to its kinetic energy.
Work-energy theorem: W_net = ΔKE = KE_final − KE_initial = ½mv_f² − ½mv_i²
This theorem states that the total work done by all forces on a body equals the change in its kinetic energy. It is particularly powerful because it relates the net force (a vector) to a scalar change in energy, bypassing the need for vector decomposition in certain problems.
Potential Energy (PE)
Stored energy arising from a body’s position in a force field or its configuration.
Gravitational Potential Energy: PE_gravity = mgh
Where m is mass, g is gravitational acceleration (9.8 m/s² near Earth’s surface), and h is the height above an arbitrarily chosen reference level. The reference level is arbitrary — setting h = 0 at ground level, at the bottom of a hill, or at infinity all produce consistent physics, merely shifting all potential energy values by a constant.
Elastic (Spring) Potential Energy: PE_spring = ½kx²
Where k is the spring constant (a measure of the spring’s stiffness, measured in N/m) and x is the displacement from the spring’s natural (unstretched, unextended) length. A larger spring constant means a stiffer spring requiring greater force for the same displacement. Hooke’s Law: F = −kx describes the restoring force that the spring exerts, directed opposite to the displacement.
General gravitational potential energy (for objects far from Earth’s surface where g changes appreciably): PE = −GMm/r
Where G is the gravitational constant (6.67 × 10⁻¹¹ N·m²/kg²), M is Earth’s mass, m is the object’s mass, and r is the distance from Earth’s center. This form is rarely required for DU-level problems but appears in advanced gravitation questions.
Conservation of Mechanical Energy
One of the most powerful principles in physics: In an isolated system where only conservative forces do work, the total mechanical energy remains constant.
Mechanical Energy = Kinetic Energy + Potential Energy = Constant
E_initial = E_final ½mv_i² + PE_i = ½mv_f² + PE_f
Conservative forces store the work they do as potential energy that can be completely recovered. Gravity and spring forces are conservative. Non-conservative forces (friction, air resistance, tension through a medium) dissipate mechanical energy as heat, sound, or deformation, and the total mechanical energy is not conserved in their presence.
Conservation of mechanical energy is equivalent to the work-energy theorem for conservative forces — both formulations yield identical predictions for isolated conservative systems.
Common energy transformations:
- Falling object: PE decreases → KE increases (PE → KE)
- Pendulum oscillation: At extremes, KE = 0, PE maximum; at bottom, KE maximum, PE minimum (PE ↔ KE continuous exchange)
- Bow and arrow: PE stored in stretched bow string → KE of released arrow
- Comet approaching Sun: PE (gravitational) converts to KE as the comet falls closer
- Roller coaster: Continuous conversion between KE and gravitational PE
Power
Power measures the rate at which work is done or energy is transferred — how quickly energy transformation occurs.
Formula: P = W/t = Energy/time
Since work and energy have the same units, power has units of energy per unit time.
SI unit: Watt (W)
- 1 Watt = 1 Joule per second = 1 J/s = 1 kg·m²/s³
- Kilowatt (kW): 1 kW = 1000 W
- Megawatt (MW): 1 MW = 10⁶ W
Other units:
- Horsepower (hp): 1 hp = 746 W (approximately ¾ kilowatt)
- erg/s: 1 erg/s = 10⁻⁷ W
Alternative power expression: P = F · v
When a constant force F acts on a body moving with velocity v in the direction of the force, power equals force times velocity. This is useful when velocity is known or measured rather than displacement and time separately. For a vehicle climbing a slope at constant speed, the engine power exactly equals the product of tractive force and velocity.
Efficiency
Real machines and processes never convert all input energy to useful output energy — some energy is always lost to heat, sound, friction, or other non-useful forms.
η = (Output energy / Input energy) × 100%
Efficiency is always less than 100% for any real device. A perpetual motion machine with 100% efficiency would create energy from nothing, violating the law of conservation of energy — such devices are physically impossible. Machines with higher efficiency waste less energy.
Collision Types and Energy
Collisions test the conservation principles and their relative applicability.
Elastic Collision: Both momentum AND kinetic energy are conserved. Objects rebound without permanent deformation or energy loss to heat. Example: billiard balls, air molecules.
Inelastic Collision: Momentum is conserved, but kinetic energy is NOT conserved — some KE converts to heat, sound, deformation energy, or other forms. Perfectly (or completely) inelastic collision: objects stick together after impact, moving with a common final velocity. Maximum kinetic energy is lost while momentum remains conserved.
Perfectly inelastic collision formula: v_f = (m₁v₁ + m₂v₂) / (m₁ + m₂)
Elastic collision in one dimension (with target initially at rest): After collision, velocities depend on the mass ratio. For equal masses, velocities simply exchange.
Key testable facts:
- In ALL collisions, momentum is always conserved (given no external forces)
- In ONLY elastic collisions, kinetic energy is conserved
- In perfectly inelastic collisions, the kinetic energy loss is maximized for given initial conditions
Spring Systems and Oscillation
Mass-spring systems undergo simple harmonic motion when displaced from equilibrium and released. The restoring force follows Hooke’s Law: F = −kx. The period (time for one complete oscillation) depends on mass and spring constant:
T = 2π√(m/k)
A stiffer spring (larger k) produces a shorter period; a larger mass produces a longer period. This formula applies only to ideal (massless) springs with the mass of the spring negligible compared to the attached mass.
Work stored in a spring: The work required to stretch or compress a spring by distance x from equilibrium equals ½kx². This equals the elastic potential energy stored in the spring: PE_spring = ½kx².
Conservative versus Non-Conservative Forces
| Property | Conservative Forces | Non-Conservative Forces |
|---|---|---|
| Work done | Path independent | Path dependent |
| Energy behavior | Work done stores as recoverable PE | Work done dissipates as heat/sound |
| Mechanical energy | Conserved | NOT conserved |
| Examples | Gravity, spring force, electrostatic | Friction, air resistance, tension with damping |
| Closed path | Zero net work | Generally non-zero net work |
Must-Remember Formulas
| Formula | Description |
|---|---|
| W = Fd cosθ | Work done by force at angle θ |
| KE = ½mv² | Kinetic energy |
| PE_gravity = mgh | Gravitational potential energy |
| PE_spring = ½kx² | Spring (elastic) potential energy |
| P = W/t | Average power |
| P = Fv | Power as force times velocity |
| η = (E_out/E_in) × 100% | Efficiency |
| v_f = (m₁v₁ + m₂v₂)/(m₁ + m₂) | Perfectly inelastic collision |
| T = 2π√(m/k) | Period of mass-spring oscillation |
| W = ½kx² | Work/stored energy in spring |
Common DU Admission Examination Patterns
- Work from force at angle: Calculate work given force magnitude, displacement, and angle
- Work-energy theorem: Given initial conditions, apply W = ΔKE to find unknown velocity or displacement
- Conservation of mechanical energy: Set KE + PE initial = KE + PE final; solve for velocity at different heights
- Power calculations: P = W/t for simple cases; P = Fv when velocity is known
- Spring problems: Combine Hooke’s Law with energy conservation — PE_spring = ½kx² converts to KE or gravitational PE
- Collision problems: Identify collision type; apply momentum conservation always, KE conservation only if elastic; determine final velocities
Examination Strategy
When problem statements give multiple quantities (mass, velocity, height, force, displacement), decide whether a force-based approach (Newton’s laws, F = ma) or an energy-based approach (work-energy theorem, energy conservation) will be simpler. Energy methods are advantageous when: (1) the path is complex or involves changing forces, (2) only initial and final states matter, or (3) friction or air resistance is negligible. Force methods are advantageous when: (1) acceleration is required, (2) time is requested, or (3) multiple forces act in different directions on the same body. For any collision problem, always write momentum conservation first — it holds regardless of elasticity. Check whether kinetic energy is also conserved only for explicitly elastic collisions. One kilowatt-hour (kWh) = 3.6 × 10⁶ J is a commercially important energy unit worth remembering.
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