Essential concepts and memory tricks for mastering Work, Energy and Power

Definition of Work and Conditions

Work is done when a force causes displacement. W = F⋅d⋅cosθ where θ is angle between force and displacement. Work is zero if: force ⊥ displacement, force = 0, or displacement = 0. Work is scalar but can be positive, negative, or zero.

Types of Energy

Kinetic Energy (KE = ½mv²): energy due to motion. Potential Energy: energy due to position or configuration. Gravitational PE = mgh, Elastic PE = ½kx². Mechanical Energy = KE + PE. All measured in joules.

Work-Energy Theorem

Net work done on an object equals change in its kinetic energy. W_net = ΔKE = KE_final - KE_initial. Very powerful for solving problems where force varies or multiple forces act. Links force and motion through energy.

Conservation of Energy

Energy cannot be created or destroyed, only transformed from one form to another. For conservative forces: KE + PE = constant. Total energy of isolated system remains constant. Foundation of many physics principles.

Conservative vs Non-conservative Forces

Conservative forces (gravity, spring force): work done is path-independent, potential energy can be defined. Non-conservative forces (friction, air resistance): work depends on path, convert mechanical energy to heat.

Power and Energy Relationship

Power is rate of doing work or transferring energy. P = W/t (average power), P = F⋅v (instantaneous power). Measured in watts (W). 1 HP = 746 W. Higher power means faster energy transfer.

Elastic Potential Energy and Springs

Energy stored in deformed elastic objects. For springs: PE = ½kx² where k is spring constant and x is compression/extension. Follows Hooke's law: F = -kx. Maximum PE at maximum deformation.

Collisions - Elastic vs Inelastic

Elastic: both momentum and kinetic energy conserved. Objects bounce off each other. Inelastic: only momentum conserved, kinetic energy decreases. Perfectly inelastic: objects stick together after collision.

All essential formulas with LaTeX equations and simple explanations

Concept Formula Meaning in Simple Words When to Use
Work by Constant Force \(\vec{F} \cdot \vec{d} = Fd\cos\theta\) Work equals force times displacement times cosine of angle between them When constant force acts over straight-line displacement
Work by Variable Force \(W = \int \vec{F} \cdot d\vec{r}\) Work is integral of force over path (area under F-x graph) When force changes with position or curved path
Kinetic Energy \(KE = \frac{1}{2}mv^2\) Kinetic energy equals half mass times velocity squared For any moving object to find energy due to motion
Kinetic Energy-Momentum Relation \(KE = \frac{p^2}{2m}\) Kinetic energy in terms of momentum p = mv When momentum is known instead of velocity
Gravitational Potential Energy \(PE_g = mgh\) Gravitational PE equals mass times gravity times height For objects in gravitational field near Earth's surface
Elastic Potential Energy \(PE_s = \frac{1}{2}kx^2\) Spring PE equals half spring constant times displacement squared For compressed or extended springs and elastic materials
Work-Energy Theorem \(W_{net} = \Delta KE = KE_f - KE_i\) Net work done equals change in kinetic energy To relate forces and motion through energy changes
Conservation of Mechanical Energy \(E = KE + PE = \text{constant}\) Total mechanical energy remains constant for conservative forces For conservative force systems (no friction/air resistance)
Average Power \(P_{avg} = \frac{W}{t}\) Average power equals work done divided by time taken When total work and time are known
Instantaneous Power \(P = \vec{F} \cdot \vec{v} = Fv\cos\theta\) Instantaneous power equals force dot product with velocity For power at any specific instant
Power-Energy Relation \(P = \frac{dE}{dt}\) Power is rate of energy transfer or transformation When energy changes continuously with time
Hooke's Law \(F = -kx\) Restoring force is proportional to displacement from equilibrium For springs and elastic materials within elastic limit
Conservation of Momentum (Collisions) \(m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2\) Total momentum before collision equals total momentum after For all types of collisions and explosions
Elastic Collision (1D) \(v_1 = \frac{(m_1-m_2)u_1 + 2m_2u_2}{m_1+m_2}\) Final velocity of first object in elastic collision For elastic collisions where both momentum and KE conserved
Coefficient of Restitution \(e = \frac{v_2 - v_1}{u_1 - u_2}\) Ratio of relative velocity of separation to approach To characterize elasticity of collisions (0 ≤ e ≤ 1)

Systematic approach to solve Work, Energy and Power problems efficiently

1

Identify the Type of Problem

Determine if it's a work problem, energy problem, power problem, or collision problem. Look for keywords: work done, energy changes, power calculations, collisions. This helps choose the right approach.

2

Draw Clear Diagrams

Sketch the situation showing forces, displacements, initial and final positions. Include coordinate system. For collisions, show before and after states. Visual representation prevents mistakes.

3

Identify Initial and Final States

Clearly define what's happening at the beginning and end. List known quantities: masses, velocities, heights, spring compressions, etc. Identify what needs to be found.

4

Choose Appropriate Method

Conservation of energy for height/speed problems. Work-energy theorem for force problems. Power formulas for time-related problems. Momentum conservation for collisions.

5

Check if Forces are Conservative

Conservative forces (gravity, springs): use energy conservation. Non-conservative forces (friction): use work-energy theorem. Mixed cases: account for energy lost to heat.

6

Set Up Equations Carefully

Write energy conservation: KE₁ + PE₁ = KE₂ + PE₂. Or work-energy: W_net = ΔKE. Include all relevant energy forms. Check signs: work can be positive or negative.

7

Solve and Check Units

Solve algebraically first, then substitute numbers. Verify units: energy in joules (J), power in watts (W). Check if answer is physically reasonable (speeds not exceeding light speed, etc.).

8

Verify Using Alternative Methods

Cross-check using different approach if possible. For collision problems, verify momentum conservation separately. Check energy balance. Ensure conservation laws are satisfied.

Avoid these common errors to improve your exam performance

Mistake How to Avoid
Getting wrong sign for work done by forces Remember: W = Fd cos θ. If force and displacement are in same direction, work is positive. If opposite directions, work is negative. Friction always does negative work on moving objects.
Confusing kinetic and potential energy KE depends on motion (½mv²), PE depends on position (mgh for gravity, ½kx² for springs). Moving objects have KE, objects at height have gravitational PE, compressed/stretched springs have elastic PE.
Misapplying conservation of energy Conservation only works for conservative forces. If friction/air resistance present, mechanical energy decreases. Use W_friction = ΔE_mechanical. Always account for all energy transformations.
Using wrong reference point for potential energy PE is relative - choose convenient reference (usually ground level = 0). What matters is ΔPE, not absolute PE. Be consistent with reference throughout problem.
Confusing power with energy Energy is capacity to do work (measured in Joules). Power is rate of energy transfer (measured in Watts = J/s). A 100W bulb uses 100J every second. P = E/t.
Wrong application of work-energy theorem Work-energy theorem: W_net = ΔKE. Use net work (sum of all forces). Only applies to kinetic energy, not total energy. For total energy changes, use W_non-conservative = ΔE_total.
Mixing up elastic and inelastic collisions Elastic: both momentum AND kinetic energy conserved. Inelastic: only momentum conserved, KE decreases. Perfectly inelastic: objects stick together. Check what's given in problem.
Forgetting to consider all forms of energy Account for all energy types: kinetic, gravitational PE, elastic PE, thermal (from friction), etc. In spring problems, don't forget both kinetic and elastic PE can be present.
Using F = kx instead of F = -kx for springs Hooke's law: F = -kx (negative sign for restoring force). Spring force opposes displacement. For PE calculation, use PE = ½kx² (always positive).
Not checking if collision formulas apply Standard elastic collision formulas only work for head-on collisions between two objects. For other cases, use conservation of momentum and energy separately.

Quick memory aids and essential information for last-minute revision

Quick Work & Energy Formulas

  • Work: W = Fd cos θ (positive if same direction)
  • Kinetic energy: KE = ½mv²
  • Gravitational PE: PE = mgh
  • Spring PE: PE = ½kx²

Energy Conservation Rules

  • Conservative forces: KE + PE = constant
  • Non-conservative forces: W_nc = ΔE_total
  • Isolated system: total energy constant
  • Energy transforms but never destroyed

Power Quick Facts

  • Average power: P = W/t
  • Instantaneous power: P = F⋅v
  • Units: 1 Watt = 1 J/s
  • 1 Horsepower = 746 Watts

Spring & Collision Points

  • Hooke's law: F = -kx (restoring force)
  • Elastic collision: momentum + KE conserved
  • Inelastic: only momentum conserved
  • Max spring PE at max compression/extension

Important Constants

  • g = 9.8 m/s² (acceleration due to gravity)
  • Work and energy units: Joule (J)
  • Power units: Watt (W) = J/s
  • 1 kWh = 3.6 × 10⁶ J (energy unit)

Problem Type Recognition

  • Height changes → use gravitational PE
  • Speed changes → use kinetic energy
  • Springs involved → include elastic PE
  • Friction present → use work-energy theorem

Exam Strategy Tips

  • Always draw energy bar charts for complex problems
  • Check energy conservation as verification
  • Watch signs carefully in work calculations
  • Use dimensional analysis to catch errors