Unit 3: Work, Energy, and Power

You’ll learn the definitions of and relationships between work, energy, and power

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3.1 Translational Kinetic Energy

Translational Kinetic Energy (TKE)

Definition

• The energy of motion an object has due to its movement from one place to another (not rotating or vibrating).

Key Characteristics

• Depends on:

  • Mass (m) – heavier objects carry more kinetic energy

  • Velocity (v) – has a greater effect due to the squared term in the formula
    Double the velocity = four times the energy

• Type:

  • A scalar quantity (no direction, only magnitude)

  • Always positive – energy can't be negative

• Frame of Reference Matters:

  • The kinetic energy of an object can vary depending on who is observing the motion

  • Think: sitting in a moving car vs. watching the car from the street

Kinetic Energy Equation

Formula: K = ½ mv²

Where:

• K = kinetic energy (in joules, J)

• m = mass (in kilograms, kg)

• v = velocity (in meters per second, m/s)

Key Relationships

• Mass and KE → Direct (Linear) Relationship

  • Double the mass → Double the kinetic energy

• Velocity and KE → Exponential (Squared) Relationship

  • Double the velocity → 4x the kinetic energy

Scalar Nature of KE

• Direction doesn’t matter
  KE is based on speed (magnitude of velocity), not the direction

  • A ball moving 3 m/s north has the same KE as one moving 3 m/s south

Always Positive

• Kinetic energy is always positive because velocity is squared in the formula: K = ½ mv²

• Squaring removes any negative sign

• An object moving at 5 m/s and one at -5 m/s have the same kinetic energy.

Same KE in Opposite Directions

• Objects moving at equal speeds in opposite directions will have identical kinetic energies.

Adding Kinetic Energies in a System

• Since kinetic energy is a scalar, to find the total energy, simply add the individual kinetic energies.

• Example: Two 1 kg balls moving toward each other at 2 m/s:

Conclusion

Direction doesn't matter in these calculations—only speed and mass determine kinetic energy.

3.2 Work

Work is the transfer of energy that occurs when a force is applied to an object and the object moves in the direction of the force.

Formula:  W = F ⋅ d ⋅ cos θ

Where:

• W = work (in joules, J)

• F = force applied (in newtons, N)

• d  = displacement of the object (in meters, m)

• θ = angle between the force vector and displacement vector

Understanding the Angle θ

• θ = 0°: Force and displacement are in the same direction
  → Maximum positive work

• θ < 90°: Force is mostly in the direction of displacement
  → Positive work

• θ = 90°: Force is perpendicular to displacement
  → No work done (since cos 90°= 0)

• θ > 90°: Force and displacement are in opposite directions
  → Negative work

Key Properties of Work

• Scalar quantity → has magnitude only, no direction

• Units: Joules (J), where 1 J = 1 N·m

Work Done by a Variable Force

• If the force changes as the object moves, work is found by calculating the area under the force vs. displacement graph (This is a graphical method)

3.3 Potential Energy

Energy Types and Their Formulas

1. Kinetic Energy (KE)

• Energy of motion

• Formula: K = ½ mv²

  • m = mass

  • v = velocity 

2. Gravitational Potential Energy (PE₉)

• Energy stored due to height above a reference point
  Formula: PE_g = mgh

  • g = acceleration due to gravity

  • h = height

3. Elastic Potential Energy (PEₑ)

• Energy stored in a stretched or compressed spring

• Formula: PE_e = ½ kx²

  • k = spring constant (stiffness)

  • x = displacement from equilibrium

Energy Transformations

Energy constantly shifts between forms depending on the situation. Key transformations include:

• Kinetic → Potential

  • A ball thrown upward slows down as it gains height

  • KE decreases, PE₉ increases

• Potential → Kinetic

  • A roller coaster descends from a hill

  • PE₉ decreases, KE increases

• Chemical → Electrical

  • In a battery-powered device, chemical energy turns into electrical energy to power circuits

Law of Conservation of Energy

• Energy cannot be created or destroyed, only transformed

• The total energy in a closed system stays constant, even as it changes form
  
  

3.4 Conservation of Energy

Key Concept:

Energy cannot be created or destroyed — only transferred or transformed.

• In a closed system, the total energy remains constant over time.

Conservative Forces (e.g., gravity, springs)

• Mechanical energy (kinetic + potential) is conserved:

KE₁ + PE₁ = KE₂ + PE₂

• No energy is lost from the system; it only changes form.

Non-Conservative Forces (e.g., friction, air resistance)

• These forces dissipate energy (usually as heat or sound).

• Mechanical energy is not conserved, but total energy still is.
  The work-energy theorem extended:

Wₙc = ΔKE + ΔPE

• Wₙc = work done by non-conservative force

Real-World Insight:
Even though energy seems to "disappear" (like from a moving object slowing down), it actually transforms (usually into heat).

3.5 Power

Definition:
Power is the rate at which work is done or energy is transferred.

Equations:

1. P = W / t

   • P = power (watts)

   • W = work (joules)

   • t = time (seconds)

2. 2. P = F × v

   • F = force (newtons)

   • v = velocity (m/s)

Units:

• Watt (W) = 1 joule per second (J/s)

Types of Power:

• Instantaneous Power: Power at a specific moment

• Average Power: Total work divided by total time