Chapter 23: How the Four Subjects Connect in Engineering Practice

The four subjects are not separate islands. An engineer uses all of them every day: mathematics to model, physics to understand the laws, chemistry to choose materials and processes, and computing to calculate, simulate, and control. This chapter shows how the ideas connect and how to practise the habits of thought that engineers use.

The modelling cycle

Engineers solve problems by building and testing models.

Define the problem clearly and decide what must be found.

Simplify and state assumptions (for example, ignore air resistance, treat a gas as ideal).

Build the model, using physical laws and mathematical equations.

Solve by algebra, calculus, or numerically with a computer.

Validate by comparing with measurements or known cases; check units and limiting cases.

Refine and communicate, stating the limits of the model.

Ideas that appear in all four subjects

An example across subjects. A capacitor of discharges through . Physics gives , so the charge decays as (Chapter 12). Mathematics solves the differential equation by separating variables (Chapter 6). Chemistry’s first-order kinetics has the same form (, Chapter 16), and radioactive decay is the same equation with a half-life (, Chapter 13). ICT can simulate it numerically, as below.

Using Python to solve a physics problem

A numerical simulation steps forward in small time intervals . For projectile motion without air resistance (Euler method):

import math

g = 9.81                     # m s^-2
v0, angle = 20.0, 30.0       # speed (m/s), launch angle (degrees)
dt = 0.001                   # time step (s)

vx = v0 * math.cos(math.radians(angle))
vy = v0 * math.sin(math.radians(angle))
x = y = t = 0.0

while y >= 0.0:
    x += vx * dt
    y += vy * dt
    vy -= g * dt
    t += dt

print(f"Range = {x:.1f} m, time of flight = {t:.2f} s")

The analytic answers are m and s (Chapter 7). The program gives values very close to these, and comparing a numerical result with the exact one is exactly how engineers validate a model. The same loop with an added drag term models a real projectile, for which no neat formula exists.

Measurement and laboratory skills

Practical work is an examination and a university skill. Key skills:

Instruments: the vernier caliper (least count commonly 0.1 mm or 0.02 mm), the micrometer screw gauge (least count 0.01 mm), stopwatch, balance, burette and pipette (volumes to cm³ and cm³ respectively, depending on the class of glassware), multimeter, and oscilloscope.

Method: repeat readings; avoid parallax; check zero errors; record in tables with units and uncertainties; control variables.

Analysis: plot linearized graphs, find gradients and intercepts, and compute uncertainties (Chapter 8).

Titrations: rinse glassware with the solution it will hold; add the titrant dropwise near the end point; repeat until titres agree (usually within 0.10 cm³); calculate concentration from the mean titre (Chapter 15).

Safety: wear eye protection; know the hazards of each chemical; never taste or smell chemicals directly; dispose of waste correctly; use electrical equipment with care; and report accidents.

From ideas to engineering

Engineering ethics. Engineers carry responsibility for safety, honesty, and the environment. Wrong data, hidden risks, or poor workmanship can harm many people. The habit begins in school: record data honestly, cite sources, and never copy another student’s work.

Common mistakes

Treating computing as a separate subject, when it multiplies the power of mathematics and physics.

Using a formula without checking that its assumptions hold.

Trusting a program’s output without checking it against a simple case.

Practice questions

A radioactive source has half-life 6 h. After how many hours does the activity fall to 1/16 of its initial value? [24 h]

Modify the Python program above to include a constant horizontal drag deceleration proportional to speed, and describe its effect on the range.

Write down one equation from each of the four subjects that has an exponential or logarithmic form.