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Energy, Work, and Power

Subject: Physics
Topic: 3
Cambridge Code: 0625

Work​

Work - Energy transferred by force

W=Fscos⁡θW = Fs\cos θ

Where:

  • F = force magnitude
  • s = displacement
  • θ = angle between force and displacement
  • Unit: Joules (J)

Special Cases​

Force parallel to motion: θ = 0°, cos(0°) = 1 W=FsW = Fs

Force perpendicular to motion: θ = 90°, cos(90°) = 0 W=0W = 0 (no work done)

Force opposite to motion: θ = 180°, cos(180°) = -1 W=−FsW = -Fs (negative work)

Kinetic Energy​

Kinetic energy - Energy of motion

Ek=12mv2E_k = \frac{1}{2}mv^2

Where:

  • m = mass
  • v = velocity

Properties:

  • Always positive
  • Zero when v = 0
  • Depends on v² (doubled speed → 4× energy)

Work-Energy Theorem​

Work done equals change in kinetic energy

Wnet=ΔEk=Ek,f−Ek,iW_{\text{net}} = \Delta E_k = E_{k,f} - E_{k,i}

Fs=12mvf2−12mvi2Fs = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2

Gravitational Potential Energy​

Gravitational PE - Energy due to height

Ep=mghE_p = mgh

Where:

  • m = mass
  • g ≈ 10 m/s²
  • h = height above reference point (usually ground)

Key points:

  • Zero at reference level (arbitrary choice)
  • Positive above reference
  • Negative below reference
  • Depends on reference frame

Change in PE​

ΔEp=mgΔh\Delta E_p = mg\Delta h

Work done against gravity to lift object = increase in PE

Elastic Potential Energy​

Energy stored in stretched/compressed spring:

Ep=12kx2E_p = \frac{1}{2}kx^2

Where:

  • k = spring constant
  • x = extension/compression from natural length

Hooke's Law: F=kxF = kx (restoring force)

Conservation of Energy​

Energy cannot be created or destroyed, only transformed

Mechanical Energy​

Etotal=Ek+Ep=constantE_{\text{total}} = E_k + E_p = \text{constant}

(When only conservative forces act)

Ek,i+Ep,i=Ek,f+Ep,fE_{k,i} + E_{p,i} = E_{k,f} + E_{p,f}

12mvi2+mghi=12mvf2+mghf\frac{1}{2}mv_i^2 + mgh_i = \frac{1}{2}mv_f^2 + mgh_f

Energy Transformations​

Examples:

  • Dropped ball: PE → KE
  • Thrown ball upward: KE → PE (then back to KE)
  • Swinging pendulum: PE ↔ KE
  • Compressing spring: Mechanical → Elastic

Non-Conservative Forces​

Non-conservative forces (like friction) dissipate energy

Ek,i+Ep,i=Ek,f+Ep,f+EdissipatedE_{k,i} + E_{p,i} = E_{k,f} + E_{p,f} + E_{\text{dissipated}}

Energy dissipated by friction: Efriction=f×d=μmgdE_{\text{friction}} = f \times d = μmgd

Work-Energy with Friction​

Wnet=Wgravity+Wfriction=ΔEkW_{\text{net}} = W_{\text{gravity}} + W_{\text{friction}} = \Delta E_k

mgh−μmgd=12mv2−0mgh - μmgd = \frac{1}{2}mv^2 - 0

Power​

Power - Rate of energy transfer

P=Wt=EtP = \frac{W}{t} = \frac{E}{t}

Unit: Watts (W) = Joules/second (J/s)

1 kilowatt (kW) = 1000 W

Power and Force​

For constant force in direction of motion: P=FvP = Fv

Where v is instantaneous velocity

Average power: Pavg=Fst=F×vavgP_{\text{avg}} = \frac{Fs}{t} = F \times v_{\text{avg}}

Efficiency​

Efficiency - Ratio of useful output to total input

η=useful outputtotal input×100%\eta = \frac{\text{useful output}}{\text{total input}} \times 100\%

Expressed as percentage (0% to 100%)

Energy Efficiency​

η=EusefulEtotal\eta = \frac{E_{\text{useful}}}{E_{\text{total}}}

Examples:

  • Electric motor: converts electrical → mechanical
  • Light bulb: converts electrical → light + heat
  • Engine: converts chemical → mechanical + heat

Power Efficiency​

η=PusefulPinputη = \frac{P_{\text{useful}}}{P_{\text{input}}}

Momentum​

Momentum - Product of mass and velocity

p=mvp = mv

Unit: kg·m/s (vector)

Rate of Change​

F=ΔpΔt=Δ(mv)tF = \frac{\Delta p}{\Delta t} = \frac{\Delta(mv)}{t}

Impulse - Change in momentum

Impulse=FΔt=Δp\text{Impulse} = F\Delta t = \Delta p

Conservation of Momentum​

Momentum conserved in closed system ptotal, before=ptotal, afterp_{\text{total, before}} = p_{\text{total, after}}

m1u1+m2u2=m1v1+m2v2m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2

Applies to:

  • Collisions
  • Explosions
  • All isolated systems

Collisions​

Elastic Collisions​

Both momentum and KE conserved

m1u1+m2u2=m1v1+m2v2m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 12m1u12+12m2u22=12m1v12+12m2v22\frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2

Inelastic Collisions​

Momentum conserved, KE not

m1u1+m2u2=m1v1+m2v2m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2

Perfectly inelastic: Objects stick together m1u1+m2u2=(m1+m2)vm_1u_1 + m_2u_2 = (m_1 + m_2)v

Key Points​

  1. Work = Force × displacement × cos(angle)
  2. KE = ½mv²
  3. PE = mgh (gravitational)
  4. PE = ½kx² (elastic)
  5. Energy conserved in isolated system
  6. Friction dissipates energy
  7. Power = energy/time = force × velocity
  8. Efficiency = useful output/total input
  9. Momentum = mv
  10. Momentum conserved in collisions

Practice Questions​

  1. Calculate work done
  2. Find kinetic energy
  3. Find potential energy
  4. Apply energy conservation
  5. Calculate power
  6. Determine efficiency
  7. Solve collision problems
  8. Apply momentum conservation
  9. Analyze energy transformations
  10. Complex energy scenarios

Revision Tips​

  • Understand work-energy theorem
  • Remember conservation laws
  • Track energy transformations
  • Use reference points for PE
  • Check units always (Joules, Watts)
  • Understand efficiency concept
  • Practice collision problems
  • Identify conservative vs non-conservative
  • Draw energy diagrams