IP Physics Notes (Upper Secondary, Year 3-4): 6) Work, Energy & Power

Quick recap -- Energy tracks the ability to do work. When forces move objects they transfer energy, and the total energy in a closed system stays constant even as it changes form.

Energy Forms & Key Equations

• Energy is the capacity to do work. SI unit: \( \pu{J} \); energy is scalar.
• Core mechanical stores:
  • Kinetic: \( E_k = \tfrac{1}{2} m v^2 \)
  • Gravitational potential: \( E_p = m g h \)
  • Elastic potential (Hooke's law springs): \( E_\text{elastic} = \tfrac{1}{2} k x^2 \)
• Other common stores (often part of the energy story): chemical, electrical, magnetic, nuclear, thermal, and radiant/light.
• Link to earlier topics: velocity can come from Chapter 2 kinematics; height can come from geometry or from Chapter 3 forces.

Principle of Conservation of Energy

• Energy cannot be created or destroyed; it can only be transferred or transformed.
• In a closed system: \( E_\text{initial} = E_\text{final} \).
• Often expressed as "loss in one store = gain in another" (allowing for energy dissipated to heat or sound when non-ideal forces act).
• Example transformations:
  • Roller coaster: \( E_p \leftrightarrow E_k \) as the cart climbs/descends.
  • Pendulum: interchange between gravitational and kinetic energy.
  • Car brakes: kinetic energy becomes thermal energy in the pads/discs.

Work Done

• Work measures the energy transferred when a force causes displacement.
• If a constant force \( F \) acts through displacement \( s \) in the same direction: \[ W = F s \]
• For angled forces use the component along displacement: \( W = F s \cos \theta \).
• Units: \( \pu{J} \). Work is scalar -- sign indicates energy added or removed.
• Positive work: force and displacement align (engine speeding up a car).
• Negative work: force opposes motion (friction slowing motion).
• Zero work: displacement is zero or force is perpendicular to motion (satellite in circular orbit).

Worked Example: Sledge on a Ramp

A ( \pu{35 kg} ) sledge is pulled ( \pu{8.0 m} ) up a slope by a rope making ( 12^\circ ) to the slope, with tension ( \pu{180 N} ). Friction opposes motion with ( \pu{65 N} ). Find the work done on the sledge by (i) the rope, (ii) friction, (iii) the net work.

1. Rope component along motion: \( W_\text{rope} = F s = 180 \times \pu{8.0 m} = \pu{1.44 \times 10^3 J} \).
2. Friction does negative work: \( W_\text{fric} = -65 \times \pu{8.0 m} = -\pu{5.2 \times 10^2 J} \).
3. Net work: \( W_\text{net} = 1.44 \times 10^3 - 5.2 \times 10^2 = \pu{9.2 \times 10^2 J} \).
4. By the work-energy theorem, the sledge gains ( \pu{9.2 \times 10^2 J} ) of kinetic plus gravitational potential energy.

Energy Conservation Applications

• Vertical drop: equate \( m g h \) to \( \tfrac{1}{2} m v^2 \) to find speed after falling.
• Spring launch: \( \tfrac{1}{2} k x^2 = \tfrac{1}{2} m v^2 \) when a compressed spring drives a cart.
• Mixed problems: include work done by friction or external forces as energy losses/gains.

Power

• Power is the rate of doing work or transferring energy. \[ P = \frac{W}{t} = \frac{\Delta E}{t} \]
• For constant speed motion with driving force \( F \): \( P = F v \).
• SI unit: watt \( \pu{W} = \pu{J.s-1} \).
• Practical note: compact answers often require using both energy and force forms depending on the data given.

Worked Example: Stair Climb Power

A student of mass ( \pu{55 kg} ) runs up a ( \pu{4.0 m} ) vertical staircase in ( \pu{3.5 s} ). Assuming negligible energy lost to heat, the power output is \[ P = \frac{m g h}{t} = \frac{55 \times \pu{9.81 m.s-2} \times \pu{4.0 m}}{\pu{3.5 s}} \approx \pu{6.16 \times 10^2 W}. \]

Efficiency

• Efficiency compares useful output to total input energy (or power): \[ \eta = \frac{E_\text{useful}}{E_\text{input}} = \frac{P_\text{useful}}{P_\text{input}} \]
• Express efficiency as a decimal or percentage. Real systems satisfy \( 0 < \eta < 1 \).
• Include dissipated energy (heat, sound) when accounting for losses.

Internal Energy (Extension)

• Internal energy sums the random kinetic and potential energies of particles in a substance.
• Heating increases particle kinetic energy; work done on the substance (e.g., compression) can raise both kinetic and potential contributions.
• During changes of state, added energy mainly increases potential energy while temperature stays constant (Chapter 9 reference).

Key Takeaways

• Track energy with consistent units and recognise when stores convert between forms.
• Work links directly to energy transfer; sign matters for gains/losses.
• Power problems demand careful handling of time or velocity data.
• Efficiency reminds you every real process sheds some energy into less useful stores; state where those losses go.