1.1 What Are Dimensions?

Dimensions are the powers to which the three fundamental quantities—mass (M), length (L), and time (T)—must be raised to represent a physical quantity. We write dimensions using square brackets, for example:

• [Mass] = M
• [Length] = L
• [Time] = T
• [Velocity] = LT⁻¹
• [Acceleration] = LT⁻²
• [Force] = MLT⁻²

A physical quantity is a measurable property of a physical system, like mass, length, time, velocity, or force. A fundamental quantity cannot be defined in terms of simpler quantities (mass, length, time). A derived quantity is obtained by combining fundamental quantities through multiplication or division.

1.2 The Principle of Dimensional Homogeneity

An equation is dimensionally correct if every term on both sides has the same dimensions.

This principle allows us to:

• Check the correctness of formulas
• Derive relationships between physical quantities
• Convert between different systems of units

1.3 Checking Formula Correctness

Example: Check if the equation $v^2 = u^2 + 2as$ is dimensionally correct, where $v$ and $u$ are velocities, $a$ is acceleration, and $s$ is distance.

Solution:

• $[v^2] = (LT^{-1})^2 = L^2T^{-2}$
• $[u^2] = L^2T^{-2}$
• $[2as] = (LT^{-2})(L) = L^2T^{-2}$

All terms have dimensions $L^2T^{-2}$, so the equation is dimensionally homogeneous.

1.4 Deriving Formulas Using Dimensional Analysis

When a physical quantity depends on other quantities, we can derive the relationship between them:

Example: The period $T$ of a simple pendulum depends on its length $l$ and acceleration due to gravity $g$. Derive the formula.

Solution:

Assume $T \propto l^a g^b$

$T = k \cdot l^a \cdot g^b$

Writing dimensions:

$[T] = [l]^a [g]^b$

$T^1 = (L)^a (LT^{-2})^b$

$M^0L^0T^1 = M^0L^{a+b}T^{-2b}$

Equating exponents:

• For T: $1 = -2b$ → $b = -\frac{1}{2}$
• For L: $0 = a + b$ → $a = \frac{1}{2}$

Therefore: $T = k\sqrt{\frac{l}{g}}$

The constant $k$ (experimentally found to be $2\pi$) cannot be determined by dimensional analysis alone.

1.5 Limitations of Dimensional Analysis

• Cannot determine dimensionless constants
• Cannot handle equations with trigonometric, logarithmic, or exponential functions
• Cannot distinguish between scalar and vector quantities
• Cannot establish relationships involving more than three fundamental quantities

2.1 Types of Errors

2.2 Types of Errors in Measurement

Absolute Error: The magnitude of the difference between measured and true values.

If measurements are $a_1, a_2, ..., a_n$, the mean value is:

$a_m = \frac{a_1 + a_2 + ... + a_n}{n}$

The absolute error in each measurement is $|a_i - a_m|$.

Mean Absolute Error:

$\Delta a_{mean} = \frac{|\Delta a_1| + |\Delta a_2| + ... + |\Delta a_n|}{n}$

The final result is written as: $a = a_m \pm \Delta a_{mean}$

Relative (Fractional) Error:

$\text{Relative error} = \frac{\Delta a_{mean}}{a_{mean}}$

Percentage Error:

$\text{Percentage error} = \frac{\Delta a_{mean}}{a_{mean}} \times 100\%$

2.3 Propagation of Errors

When calculating derived quantities, errors propagate according to these rules:

For sum or difference: $x = a \pm b$ $\Delta x = \Delta a + \Delta b$

For product or quotient: $x = a \cdot b$ or $x = \frac{a}{b}$ $\frac{\Delta x}{x} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$

For powers: $x = a^n$ $\frac{\Delta x}{x} = n \cdot \frac{\Delta a}{a}$

Worked Example:

The period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{l}{g}}$. If length $l$ has a 2% error and period $T$ has a 3% error, find the percentage error in $g$.

Solution:

From $g = 4\pi^2 \frac{l}{T^2}$:

$\frac{\Delta g}{g} = \frac{\Delta l}{l} + 2\frac{\Delta T}{T}$

$\frac{\Delta g}{g} = 2\% + 2(3\%) = 8\%$

A measurement can be precise without being accurate, and vice versa.

Improving Accuracy:

• Use instruments with higher precision
• Avoid parallax errors (read scale at right angles)
• Repeat measurements and take averages
• Control environmental conditions

The least count is the smallest measurement an instrument can read reliably, equal to one scale division.

• Metre rule: 1 mm (least count = 1 mm)
• Vernier calipers: 0.1 mm
• Micrometer screw gauge: 0.01 mm
• Stopwatch: typically 0.1 s or 1 s

The minimum uncertainty in a direct reading is usually half the least count.

Problem: A student measures the length of a pendulum string as 85.0 cm using a metre rule, then measures the time for 10 oscillations as 18.4 s using a stopwatch.

Given:

• Least count of metre rule = 1 mm
• Least count of stopwatch = 0.1 s
• Systematic error in stopwatch = +0.2 s

Find: (a) Absolute error in length (b) Absolute error in time (c) Percentage error in the calculated value of $g$

Solution:

(a) Absolute error in length = half least count = ±0.05 cm

(b) Absolute error in time = half least count + systematic error $= \pm 0.05 s + 0.2 s = \pm 0.25 s$

(c) From $T = 2\pi\sqrt{\frac{l}{g}}$, we get $g = 4\pi^2 \frac{l}{T^2}$

Percentage error in $g$: $\frac{\Delta g}{g} = \frac{\Delta l}{l} + 2\frac{\Delta T}{T}$

$\frac{\Delta g}{g} = \frac{0.05}{85.0} \times 100\% + 2 \times \frac{0.25}{1.84} \times 100\%$

$\frac{\Delta g}{g} = 0.06\% + 27.2\% = 27.26\%$

1. Dimensional analysis uses M, L, T to check formula correctness and derive relationships
2. Systematic errors are consistent; random errors vary unpredictably
3. Absolute error = mean absolute error ± uncertainty
4. Relative error = absolute error ÷ measured value
5. When multiplying/dividing quantities, add relative errors
6. When raising to power $n$, multiply relative error by $n$
7. Precision = repeatability; Accuracy = closeness to true value

In Tanzania, farmers and traders at local markets use measuring principles when weighing produce like maize, beans, or rice using balance scales. Understanding uncertainties helps a maize farmer at the Mbeya market determine whether a 50 kg sack actually weighs close to 50 kg, or whether the scale has systematic error that could lead to financial loss when selling by weight. Similarly, constructors in Dar es Salaam must account for measurement uncertainties when calculating cement, sand, and aggregate proportions for concrete mixing—small percentage errors in volume can significantly affect structural strength and cost.