The physics behind engineering and the physical sciences: motion, forces, energy, momentum, torque, circuits, induction, waves and heat, with Python exercises and a home pendulum lab.

Physics for Engineering & Science

## Why this matters

Anything that is pulled back toward a resting position tends to oscillate: a mass on a spring, a pendulum, a building in an earthquake, a car on its suspension, the current in a tuned radio circuit. And when an oscillation spreads out through a material or through space, it becomes a **wave**: sound, light, radio, ocean swell, seismic waves. Engineers design to avoid unwanted vibration; physicists and chemists use waves (light, X-rays, ultrasound) to measure the world. One small equation, v = f·λ, connects most of it.

## Describing an oscillation

- **Amplitude** A: the largest distance from the rest position.
- **Period** T (s): the time for one full cycle.
- **Frequency** f (hertz, Hz): cycles per second. f = 1 / T.
- **Angular frequency** ω = 2π·f (rad/s).

A swing that takes 2.0 s for a full back-and-forth has f = 0.5 Hz. Household mains alternates at 50 or 60 Hz, so one cycle lasts 20 or about 16.7 ms.

## Simple harmonic motion

When the restoring force is proportional to the displacement (F = −k·x, Hooke's law for a spring), the motion is **simple harmonic**: the position follows a sine wave,

```text
x(t) = A · sin(ω·t)
```

For a mass m on a spring of stiffness k:

```text
T = 2π · √(m / k)
```

A 0.50 kg mass on a 200 N/m spring: T = 2π × √(0.0025) = 2π × 0.050 = 0.314 s, so f ≈ 3.2 Hz. The amplitude does not appear: big or small, the oscillation takes the same time.

For a simple pendulum of length L, for small swings (under about 15°):

```text
T = 2π · √(L / g)
```

A 1.00 m pendulum has T = 2π × √(1.00 / 9.81) = 2.01 s. Notice that the mass is not in the formula. You will use this in the lab lesson to measure g.

## Resonance

Every oscillating system has a **natural frequency**. Push it periodically at that frequency and each push adds to the motion, so the amplitude can grow large: that is **resonance**. You use it to pump a playground swing; tuned radio circuits use it to pick one station. Engineers must keep machinery, bridges and buildings away from resonating with wind, footsteps, engines or earthquakes, and that analysis is specialist work that is reviewed by licensed engineers.

## Waves

A wave carries energy from place to place without carrying the material along with it. A cork on a pond bobs up and down as ripples pass; it does not travel with them.

- **Transverse waves:** the medium moves perpendicular to the wave's direction (a wave on a rope, light).
- **Longitudinal waves:** the medium moves back and forth along the direction of travel (sound, which is a series of compressions and rarefactions in air).

The **wavelength** λ (lambda) is the distance between two neighbouring crests. In one period T the wave moves forward one wavelength, so its speed is

```text
v = λ / T = f · λ
```

The speed is set by the medium, not by the source. Sound travels at about 343 m/s in air at 20 °C, about 1,480 m/s in water and around 5,000–6,000 m/s in steel. Light in a vacuum travels at c = 3.00 × 10⁸ m/s.

## Worked examples

**Concert A.** A tuning fork vibrates at 440 Hz in air:

```text
λ = v / f = 343 / 440 = 0.780 m
```

**FM radio.** A station at 100 MHz (1.00 × 10⁸ Hz):

```text
λ = c / f = 3.00 × 10⁸ / 1.00 × 10⁸ = 3.00 m
```

**Wi-Fi.** At 2.4 GHz, λ = 3.00 × 10⁸ / 2.4 × 10⁹ = 0.125 m, about 12.5 cm. Antenna sizes are often a simple fraction of the wavelength, which is why Wi-Fi antennas are a few centimetres long.

**Same frequency, new medium.** When sound goes from air into water the frequency stays the same (it is set by the source), but the speed rises, so the wavelength grows: 440 Hz in water gives 1480 / 440 = 3.36 m.

## Superposition

When two waves overlap, their displacements add. Crest on crest gives a bigger wave (**constructive interference**); crest on trough cancels (**destructive interference**). Noise-cancelling headphones play an inverted copy of incoming sound to cancel it.

## In code

```python
import math

def spring_period(mass, k):
    return 2 * math.pi * math.sqrt(mass / k)

print(round(spring_period(0.50, 200), 3))   # 0.314
```

## Common mistakes

- **Mixing up period and frequency.** They are reciprocals: 4 Hz means a 0.25 s period.
- **Forgetting the unit prefixes.** MHz is 10⁶ Hz and GHz is 10⁹ Hz.
- **Thinking a higher frequency travels faster.** In air, all audible sound travels at about the same speed; higher frequency just means shorter wavelength.
- **Using the pendulum formula for big swings.** Beyond about 15° the period grows slightly.

## Quick recap

- f = 1 / T and ω = 2πf.
- Spring: T = 2π√(m/k). Pendulum (small swings): T = 2π√(L/g).
- Wave speed v = f·λ; the medium sets v, the source sets f.
- Sound in air ≈ 343 m/s; light ≈ 3.00 × 10⁸ m/s.

## Try it

In the **Do** tab, write a wavelength function and test it on sound, radio and light.


Describe oscillations and waves, and relate frequency, wavelength and speed with v = f·λ.

Waves and oscillations

Traveling Waves: Crash Course Physics #17

Simple Harmonic Motion: Crash Course Physics #16

## Why this matters

Heat is energy on the move, and almost every engineered system has to manage it: engines and power plants turn heat into work, electronics must shed heat to survive, buildings are insulated against it, and chemical reactors are held at precise temperatures. Chemists use heat measurements (calorimetry) to find reaction energies, and materials scientists heat-treat metals to change their properties. This lesson gives you the core calculation, Q = m·c·ΔT, plus the three ways heat travels.

## Temperature and heat are different

- **Temperature** measures how energetic the random motion of a material's particles is, on average. It is measured in degrees Celsius (°C) or kelvin (K).
- **Heat** Q is energy transferred because of a temperature difference, always flowing from hotter to colder. It is measured in joules.

A bathtub of warm water and a cup of boiling water: the cup is hotter (higher temperature), but the bathtub holds far more thermal energy and can melt far more ice.

The kelvin scale starts at absolute zero:

```text
T(K) = T(°C) + 273.15
```

A **change** of 1 °C is exactly the same as a change of 1 K, so in ΔT you can use either. Use kelvin whenever a formula uses the temperature itself (as in the gas laws).

## Specific heat capacity

Different materials need different amounts of energy to warm up. The **specific heat capacity** c is the energy needed to raise 1 kg by 1 K:

```text
Q = m · c · ΔT
```

| material | c, J/(kg·K), approximate |
|---|---|
| water (liquid) | 4,186 |
| ice | 2,100 |
| aluminium | 900 |
| iron / steel | 450 |
| copper | 385 |

Water's unusually high c is why oceans moderate climate and why water is used as a coolant in engines and power plants.

## Worked example 1: boiling water in a kettle

How much heat raises 1.5 kg of water (1.5 litres) from 20 °C to 100 °C?

```text
Q = 1.5 × 4186 × (100 − 20) = 502,320 J ≈ 502 kJ
```

A 2,000 W kettle supplies 2,000 J per second. If all of it went into the water:

```text
t = Q / P = 502,320 / 2,000 = 251 s ≈ 4.2 minutes
```

Real kettles take a bit longer because some heat warms the kettle itself and escapes to the room. The ratio of useful heat to energy supplied is the **efficiency**.

## Worked example 2: mixing (calorimetry)

Pour 0.20 kg of water at 80 °C into 0.30 kg of water at 20 °C in an insulated cup. If no heat escapes, the heat lost by the hot water equals the heat gained by the cold water:

```text
0.20 × 4186 × (80 − T) = 0.30 × 4186 × (T − 20)
16 − 0.20·T = 0.30·T − 6
T = 22 / 0.50 = 44 °C
```

With the same material on both sides, c cancels and the answer is a mass-weighted average. With different materials, keep each c in the equation.

## Phase changes

While a substance melts or boils, heat goes into breaking bonds instead of raising the temperature. The energy per kilogram is the **latent heat** L:

```text
Q = m · L
```

For water, the latent heat of fusion (melting) is about 334 kJ/kg and of vaporisation (boiling) about 2,260 kJ/kg. Boiling 1.5 kg of water away completely would take another 3.4 MJ, almost seven times as much as heating it from 20 °C to 100 °C. That huge number is why steam burns are so severe and why evaporative cooling works.

## How heat travels

1. **Conduction** through a solid, particle to particle. For a flat slab:

```text
P = k · A · ΔT / d
```

k is the thermal conductivity in W/(m·K), A the area, d the thickness. A 10 m² wall section with 0.10 m of insulation (k ≈ 0.04 W/(m·K)) and a 20 K difference loses about 0.04 × 10 × 20 / 0.10 = 80 W. Copper (k ≈ 400) conducts ten thousand times better, which is why heat sinks are metal.

2. **Convection:** a moving fluid carries heat with it. Warm air rising from a radiator, a fan cooling a CPU.

3. **Radiation:** every object emits electromagnetic radiation, mostly infrared at everyday temperatures. It needs no medium, which is how sunlight reaches Earth.

Real building and equipment calculations combine all three and use code-approved methods and material data; this lesson gives the physics, not a design procedure.

## In code

```python
def mix_temperature(m1, t1, m2, t2):
    """Final temperature when two masses of water mix with no heat loss."""
    return (m1 * t1 + m2 * t2) / (m1 + m2)

print(mix_temperature(0.20, 80, 0.30, 20))   # 44.0
```

## Common mistakes

- **Using grams with c in J/(kg·K).** 250 g is 0.250 kg.
- **Confusing heat and temperature.** More mass at the same temperature holds more energy.
- **Forgetting phase changes.** Heating ice at −10 °C to water at 20 °C has three steps: warm the ice, melt it, warm the water.
- **Using °C in a formula that needs kelvin.** ΔT is fine in °C; absolute temperatures in gas laws are not.

## Quick recap

- Heat is energy transferred by a temperature difference; temperature is not heat.
- Q = m·c·ΔT; water's c ≈ 4,186 J/(kg·K).
- Phase changes: Q = m·L, with no temperature change.
- Heat travels by conduction, convection and radiation.

## Try it

In the **Do** tab, write the heat equation and use it to work out how long a kettle takes to boil.


Distinguish heat from temperature and compute heat transfer with specific heat, latent heat and conduction.

Heat and temperature

The Physics of Heat: Crash Course Physics #22

Specific heat capacity | Khan Academy

Tie the weight to the string and tape the other end firmly to the edge of a sturdy table or a door frame so the weight hangs freely without touching anything.

Measure the length L from the point where the string is fixed to the centre of the weight, in metres, to the nearest half centimetre. Aim for about 0.8–1.0 m.

Pull the weight a short way to one side (about 15–20 cm for a 1 m string, so the angle stays under 15°) and release it without pushing.

Start the stopwatch as the weight passes the lowest point and count that pass as zero. Stop after 20 complete swings (back to the same point moving the same way). Record the time.

Repeat the timing two more times. Average the three times and divide by 20 to get the period T.

Calculate g = 4π² × L / T². Then estimate the uncertainty: δg/g ≈ δL/L + 2 × δT/T, using δL = 0.005 m and δT = 0.2 s / 20 = 0.01 s.

Shorten the string to about 0.6 m and then about 0.4 m, and repeat steps 2–5 for each length.

Plot T² (vertical) against L (horizontal) on paper or in a spreadsheet, draw the best straight line through the points and the origin, and find g = 4π² / slope.

Compare both of your results with the accepted value of about 9.81 m/s². Write down whether it lies inside your uncertainty range and list the two largest sources of error.

## Why this matters

Physics is an experimental science. Every equation in this course was accepted because measurements agreed with it, and every engineer and scientist spends part of their career collecting data, estimating its uncertainty and deciding whether it supports a conclusion. This lab uses a string, a small weight and a stopwatch to measure the acceleration due to gravity, g, to within a couple of percent. The method (plan, measure several times, reduce error, compare with a model) is the same one used in research and test labs.

## The model

From the waves lesson, a simple pendulum of length L swinging through small angles has period

```text
T = 2π · √(L / g)
```

Rearranged for g:

```text
g = 4π² · L / T²
```

So measuring L and T gives g. Three assumptions hide in the model, and the procedure is designed to respect them:

1. **Small angles.** The formula is very accurate below about 15°. Pull the weight only a short way to the side (for a 1 m string, about 20 cm).
2. **A point mass on a light string.** Use a small, dense weight such as a large metal nut, a bundle of keys or a fishing sinker, on thin string or dental floss. Measure L to the **centre** of the weight.
3. **No friction or drag.** A dense weight keeps air drag small, and the swings should die away only slowly.

## Measuring the period well

Your reaction time when starting and stopping a stopwatch is roughly 0.1–0.2 s. That error is huge compared with a single period of about 2 s. The fix is to time **many** swings and divide:

```text
20 swings in 35.9 s   →   T = 35.9 / 20 = 1.795 s
```

The reaction-time error is now shared across 20 swings, so it drops to about 0.01 s per period. Start timing as the weight passes the lowest point (where it moves fastest, so the moment is easiest to judge) and count "zero" at that first pass.

Repeat each timing three times. If the three values differ by more than a few tenths of a second, you probably miscounted.

## Worked example

L = 0.800 m and the average of three timings gives T = 1.795 s:

```text
g = 4π² × 0.800 / 1.795² = 31.58 / 3.222 = 9.80 m/s²
```

## How good is the answer?

A result without an uncertainty is only half an answer. For a formula like g = 4π²L/T², the relative uncertainties add, with the period counted twice because it is squared:

```text
δg / g ≈ δL / L + 2 · δT / T
```

If you can measure L to ±0.5 cm (0.005 m) and T to ±0.01 s:

```text
δL / L = 0.005 / 0.800 = 0.6%
2·δT/T = 2 × 0.01 / 1.795 = 1.1%
total  ≈ 1.7%   →   g = 9.80 ± 0.17 m/s²
```

The accepted value, about 9.81 m/s² (it varies slightly with latitude and altitude), lies well inside that range, so the experiment agrees with the model. If yours does not, look for a cause: a wrong length (measured to the knot instead of the centre of the weight), large swings, or a miscount.

## A better method: several lengths

Measuring at one length tests one point. Measuring at three to five lengths and plotting **T² against L** is stronger. The model predicts a straight line through the origin with slope 4π²/g, so

```text
g = 4π² / slope
```

A line fitted through several points averages out random errors, and a curve or an offset reveals systematic ones. That is the habit to keep: test a model across a range, not at a single point. You can fit the slope by hand on graph paper, in a spreadsheet, or with a few lines of Python later in the Scientific Python course.

## Optional: phone sensors

The free **phyphox** app (from RWTH Aachen University) turns a phone's accelerometer, gyroscope and microphone into lab instruments and includes a pendulum experiment. Only use the phone as the swinging weight if it is securely held in a bag hanging just above a bed or cushion, so a slip cannot damage it. Otherwise, use the phone simply as a stopwatch.

## Safety

This lab is safe at home if you keep it simple: hang the string from a sturdy table edge, shelf bracket or door frame you can reach from the floor (no standing on chairs), keep the swing small, and keep it clear of glass, pets and young children.

## Where this leads

This course gives you the physics vocabulary that engineering mechanics, thermodynamics, circuits and computational physics build on. The careers it serves require far more than any online course can provide. Engineers usually need a bachelor's degree in engineering (an ABET-accredited program is the standard route to licensure), and those whose work affects public safety pass the FE and PE exams and work under or as a licensed Professional Engineer: see NCEES (https://ncees.org). Physicists, chemists and materials scientists need at least a bachelor's degree with real laboratory coursework, and research roles usually need a graduate degree. Checked on 2026-10-23. Real designs always need review by a licensed engineer; the calculations in this course build intuition, not sign-off.

## Quick recap

- g = 4π²L/T²; measure L to the centre of the weight and keep swings small.
- Time many swings to shrink reaction-time error.
- Relative uncertainty: δg/g ≈ δL/L + 2δT/T.
- Several lengths and a T²-versus-L plot test the model better than one point.

## Try it

Open the **Do** tab and follow the steps to measure g at home.


Measure the acceleration due to gravity with a home-made pendulum and estimate the uncertainty of your result.

Physics lab at home

Smartphone-Experiment: Pendulum (en)

Period of a Pendulum | Simple harmonic motion and rotational motion | AP Physics 1 | Khan Academy

Oscillations and waves, heat and specific heat, and a home pendulum lab that measures g with an uncertainty estimate.