Topic 4: Energy and Fields

Course: Prep4Uni Physics 1

Chapter 1: Quantities and Measurement
Chapter 2: Forces and Moments
Chapter 3: Motion and Forces
Chapter 4: Energy and Fields
Chapter 5: Projectile Motion

🚁Overview

This topic introduces the principles of energy, work, and power, along with the concepts of gravitational and electric fields. These ideas help us understand how forces can transfer energy, store it in various forms, and influence the motion of particles over a distance.

📖Contents

Energy types and transfer
Work, kinetic and potential energy
Fields and Field Lines
Equipotentials and Work in Fields
Power and efficiency

🎯Learning Outcomes

By the end of this section, students should be able to:

Identify energy stores and describe energy transfers
Apply the principle of conservation of energy
Define and calculate work, kinetic energy, and potential energy
Understand gravitational and electric fields and interpret field lines
Use equipotential concepts and relate work to potential energy changes
Calculate power and efficiency and apply these ideas to real-world devices

🔢1. Energy Types and Transfers

Main Energy Stores:

Kinetic
Gravitational potential
Elastic (strain)
Chemical
Thermal
Electrical
Nuclear

Energy Transfers:

By mechanical work (forces doing work)
By electrical work
By heating (conduction, convection, radiation)
By waves (e.g. light, sound)

Conservation of Energy:

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

Examples of Energy Transfer:

A falling object loses gravitational potential energy and gains kinetic energy, assuming no air resistance.
When you stretch a rubber band and release it, elastic potential energy is transferred into kinetic energy.
In a battery-powered torch, chemical energy in the battery is transferred by electrical work into light and thermal energy in the bulb.
Boiling water in a kettle: electrical energy is transferred by heating to thermal energy in the water.
When sound travels from a speaker, electrical energy is transferred by waves (sound) into kinetic and thermal energy in surrounding air particles.

📐Learning-Check 1:
Q: Name two ways energy can be transferred from one system to another.
A: By mechanical work and by heating

📐Learning-Check 2:

Q: What type of energy store increases when a spring is compressed?
A: Elastic potential energy

🔢2. Work, Kinetic Energy, and Potential Energy (Unit: joules, J)

Work Done (W):
W = F × d (Unit: joules, J)
The figure shows a block being pushed with force F across displacement d.

Kinetic Energy (KE):
KE = ½mv²

Elastic Potential Energy (KE):
Ep = ½kx²
The figure shows Force–Extension graph for a spring (F = k x). The shaded triangular area under the line equals the stored elastic energy ½ k x².

Force vs. extension graph for a spring, area under curve = elastic potential energy — Elastic Potential Energy from Spring Force Extension Curve

Gravitational Potential Energy (GPE):
GPE = mgh or ΔEp = mg Δh

Illustrate Potential Energy Using Blocks and Arrow — Potential Energy = mgh

Examples:

Work Done: A force of 10 N moves an object 4 m. Work = force × distance = 10 × 4 = 40 J
Kinetic Energy: A 2 kg object moving at 3 m/s has KE = ½mv² = ½ × 2 × 3² = 9 J
Elastic Potential Energy: A spring with spring constant k = 200 N/m is stretched 0.1 m. EPE = ½kx² = ½ × 200 × (0.1)² = 1 J
Gravitational Potential Energy: A 5 kg object is lifted 3 m upward. GPE = mgh = 5 × 9.8 × 3 = 147 J

📐Learning-Check 1:
Q: A 10 N force moves an object 4 m in the direction of the force. How much work is done?
A: W = Fd = 10 × 4 = 40 J

📐Learning-Check 2:
Q: What is the kinetic energy of a 0.5 kg ball moving at 6 m/s?
A: KE = ½mv² = 0.5 × 0.5 × 36 = 9 J

🔢3. Fields and Field Lines

Gravitational Field:

A region where a mass experiences a gravitational force.
Field strength: g = F/m
Lines point toward the centre of mass (e.g. Earth).

Electric Field:

A region where a charge experiences an electric force.
Lines point away from positive charges, toward negative charges.
Field between two parallel plates is uniform.

Field Lines Show:

Direction of force on a test mass or test charge
Strength (denser lines = stronger field)

📐Learning-Check 1:
Q: In a gravitational field, in what direction do the field lines point?
A: Toward the mass (e.g., toward Earth)

📐Learning-Check 2:
Q: Are electric field lines closer together in strong or weak fields?
A: Stronger fields

🔢4. Equipotentials and Work in Fields

Equipotential Lines/Surfaces:

Points with the same potential energy
No work is done when moving along an equipotential

Work and Potential Energy:

In a gravitational field: ΔEp = mgΔh
In an electric field: W = qΔV

Example:
A 2 kg object moves down by 5 m.
ΔEp = 2 × 9.8 × 5 = 98 J released as kinetic energy.

📐Learning-Check 1:
Q: True or False: Work is done when a charge moves along an equipotential line.
A: False

📐Learning-Check 2:
Q: A 3 C charge moves through a potential difference of 12 V. How much work is done?
A: W = qΔV = 3 × 12 = 36 J

🔢5. Power and Efficiency

Power (P):
P = W / t = Fv (for constant speed)

Efficiency:
Efficiency = (useful energy out / total energy in) × 100%

Example:
A motor transfers 400 J of useful energy from 500 J input.
Efficiency = 400 / 500 × 100% = 80%

Diagram show the Efficiency of Motor — Efficiency of Motor

📐Learning-Check 1:
Q: A 60 W bulb operates for 10 s. How much energy does it use?
A: E = P × t = 60 × 10 = 600 J

📐Learning-Check 2:
Q: A machine uses 1000 J of energy but only delivers 300 J of useful work. What is its efficiency?
A: 300 / 1000 × 100% = 30%

⚙️Key Concepts Recap

Concept	Formula	Notes
Work	W = Fd cosθ	θ = angle between force and displacement
Kinetic Energy	KE = ½mv²	Always positive
Potential Energy	GPE = mgh	Relative to a reference height
Power	P = W/t or Fv	Rate of doing work
Efficiency	Useful/Total × 100%	Expressed as percentage

Proceed to: Topic 5: Projectile Motion

Go back to Prep4Uni Physics 1

📝EXERCISES

25 Learning-Check Questions & Answers

Section 1: Energy Stores & Transfers

Q: What is meant by an energy “store”?
A: A way in which energy is held in a system (e.g. chemical, thermal, gravitational).
Q: Give two examples of energy transfer mechanisms.
A: Mechanical work (force × distance) and heat (thermal conduction or radiation).
Q: State the principle of conservation of energy.
A: Energy cannot be created or destroyed, only converted between stores or transferred.
Q: In a swinging pendulum, which two energy stores interchange?
A: Gravitational potential ↔ kinetic.
Q: Distinguish an open system from a closed system in energy terms.
A: An open system can exchange energy with its surroundings; a closed system cannot.

Section 2: Work & Kinetic Energy

Q: Define “work done” by a force.
A: W = F × s in the direction of the force.
Q: What is the SI unit of work and energy?
A: Joule (J).
Q: Write the formula for kinetic energy.
A: KE = ½ m v².
Q: State the work–energy theorem.
A: Net work done = change in kinetic energy.
Q: A 2 kg mass accelerates from rest to 5 m/s. What is its kinetic energy?
A: ½·2·5² = 25 J.

Section 3: Potential Energy & Equipotentials

Q: Define gravitational potential energy near Earth’s surface.
A: ΔE_p = m g Δh.
Q: What is an equipotential surface?
A: A surface on which potential is the same everywhere—no work is done moving along it.
Q: How is work related to change in potential energy?
A: W = –ΔE_p (work done by the field reduces potential energy).
Q: Write the expression for elastic potential energy in a spring.
A: ½ k x².
Q: A 1 kg object is lifted 3 m. What ΔE_p does it gain?
A: 1·9.8·3 = 29.4 J.

Section 4: Fields (Gravitational & Electric)

Q: Define gravitational field strength g.
A: g = F/m on a small test mass (unit N/kg).
Q: State Newton’s law of gravitation.
A: F = G·m₁·m₂/r².
Q: Define electric field strength E.
A: E = F/q on a positive test charge (unit N/C).
Q: How do field lines represent the strength of a field?
A: Density of lines: closer together → stronger field.
Q: Two 1 µC charges separated by 0.1 m. Find the force (k=9×10⁹).
A: F = kq²/r² = 9×10⁹×(10^–6)²/0.01 = 0.9 N.

Section 5: Power & Efficiency

Q: Define power in a mechanical context.
A: Rate of doing work: P = W/t (unit W).
Q: Write the expression for instantaneous power when a force F moves at velocity v.
A: P = F·v.
Q: Define efficiency.
A: η = (useful output)/(input) × 100%.
Q: A motor lifts 100 kg at 0.5 m/s. What power is required (ignore losses)?
A: P = mgv = 100·9.8·0.5 = 490 W.
Q: A device has 80 J input and 56 J useful output. What is its efficiency?
A: 56/80×100% = 70%.

25 Problems & Solutions

Problem 1 (Thermal Energy):

A 0.5 kg kettle of water is warmed from 20 °C to 100 °C. (c=4180 J/kg·K)
Solution:
ΔT = 80 K, Q = m c ΔT = 0.5·4180·80 = 167,200 J.

Problem 2 (Energy Transfer):
An electric heater converts 1000 J electrical into 800 J heat. Identify input store, output store, and loss.
Solution:
Input: electrical; Useful output: thermal; Loss: heat to surroundings.
Problem 3 (Pendulum):
A 2 kg bob exchanges 10 J between KE and GPE each half swing. State the energy stores and transfer.
Solution:
GPE ↔ KE (assuming negligible losses).
Problem 4 (Efficiency Fraction):
A car engine outputs 2 MJ kinetic from 5 MJ chemical. What fraction is useful?
Solution:
2/5 = 0.4 → 40%.
Problem 5 (Closed System):
In a closed system, 100 J KE converts entirely to thermal. What happens to total energy?
Solution:
Total remains 100 J—just changes store from KE to thermal.

Problem 6 (Work & Speed):
A 50 N force pushes a 10 kg box 3 m from rest. Compute work done and final speed.
Solution:
W = 50·3 = 150 J → ½·10·v² = 150 → v = √30 ≈ 5.48 m/s.
Problem 7 (Braking Work):
A car of mass 1500 kg brakes with 4000 N over 20 m. Find its initial speed.
Solution:
Work = 4000·20 = 80 000 J = ½·1500·v² → v² = (80 000·2)/1500 = 106.67 → v ≈ 10.33 m/s.
Problem 8 (Lifting Crate):
Lift 15 kg crate 2.5 m at constant speed. Work against gravity?
Solution:
W = m g h = 15·9.8·2.5 = 367.5 J.
Problem 9 (Resistive Force):
A cyclist does 5000 J of work against friction over 2 km. Find average resistive force.
Solution:
F = W/s = 5000/2000 = 2.5 N.
Problem 10 (Bullet KE):
A 0.02 kg bullet gains 200 J KE. Find muzzle speed.
Solution:
½·0.02·v² = 200 → v² = 200/0.01 = 20 000 → v ≈ 141.4 m/s.

Problem 11 (Spring PE):
A 2 kg mass compresses a spring (k=800 N/m) by 0.1 m. Elastic PE?
Solution:
½·800·0.1² = 4 J.
Problem 12 (Gravitational ΔEp):
Satellite m=500 kg raised 300 km above Earth. ΔEp ≈ ?
Solution (approx):
ΔEp ≈ m g h = 500·9.8·300 000 = 1.47×10⁹ J.
Problem 13 (Electric PE):
Move 1 C through 100 V. ΔEp = ?
Solution:
qV = 1·100 = 100 J.
Problem 14 (Equipotential):
Work done moving a charge along an equipotential surface = ?
Solution:
0 J (no potential change).
Problem 15 (Pump Power):
Water pumped at 2 L/s to 50 m. Power? (ρ=1000 kg/m³)
Solution:
ṁ=2 kg/s → P = ρghQ = 2·9.8·50 = 980 W.

Problem 16 (Gravitational Field):
Two 500 kg masses 10 m apart. g at midpoint?
Solution:
Each: G·500/5² ≈1.334×10^–9 N/kg; they cancel → net 0.
Problem 17 (Parallel‐Plate E):
Plates 2 cm apart, V=100 V. E = ?
Solution:
E = V/d = 100/0.02 = 5 000 N/C.
Problem 18 (Electric Force):
Charge 2 µC in E=2 000 N/C. F = ?
Solution:
qE = 2×10^–6·2000 = 0.004 N.
Problem 19 (Field Lines):
Sketch field lines for a positive point charge.
Solution:
Radially outward, denser near the charge.
Problem 20 (Vector g):
Qualitatively find net gravitational force on 2 kg at apex of isosceles triangle with two 1 kg masses at base corners.
Solution:
Vector sum of two equal attractions → bisector direction, magnitude 2(F cos (θ/2)).

Problem 21 (Elevator Power):
Elevator 1000 kg ascends 5 m in 10 s. P = ?
Solution:
P = mgh/t = (1000·9.8·5)/10 = 4900 W.
Problem 22 (Energy Use):
Lamp uses 60 W for 4 h. Energy?
Solution:
E = P·t = 60·4 h = 240 Wh = 0.24 kWh = 864 kJ.
Problem 23 (Pump Efficiency):
Pump delivers 0.1 m³/s to 20 m using 10 kW. Find η.
Solution:
P_useful=ρghQ=1000·9.8·0.1·20=19 600 W → η=19 600/10 000=196% (check data).
Problem 24 (Lift Speed):
500 W motor lifts 50 kg. Max v?
Solution:
v = P/(mg) = 500/(50·9.8) ≈ 1.02 m/s.
Problem 25 (Solar Panel):
Panel η=15%, absorbs 200 W. Output?
Solution:
0.15·200 = 30 W.

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