MHT-CET Physics · Units and Measurement

Units and Errors in Measurement

A repeated measurement is reported as its mean ± its mean absolute error; absolute errors add in a sum or a difference, and percentage errors add in a product or a quotient, each multiplied by the power its quantity is raised to.

All Units and Measurement notes MHT-CET Physics strategy

Why this matters

14 PYQs, none HARD. Ten ask for the percentage error in a density, a pressure, a kinetic energy, g from a pendulum, or a general product of powers. Four are a mean with its error, the error in a temperature rise, which errors are random, and the unit of L/R. Two cards.

Concept 1 of 2: Means, Absolute Errors and Kinds of Error

Several readings are reported as their mean, with the mean of their absolute deviations as the error: 30, 32, 35, 35 s give 33 ± 2 s. In a sum or a difference the absolute errors ADD, never subtract: (82.3 ± 0.3) − (38.6 ± 0.2) = 43.7 ± 0.5. Systematic errors push every reading the same way (a badly calibrated thermometer, a zero error); random errors scatter them (a fluctuating supply). Units follow from formulas: L/R is the time constant of an LR circuit, measured in seconds.

Definition

Mean $\bar{x}$ ; mean absolute error $\overline{|x_i - \bar{x}|}$ .
Sum or difference: $\Delta Z = \Delta A + \Delta B$ .
Systematic: calibration, zero error. Random: unpredictable fluctuations. Gross: misreadings.
$\dfrac{L}{R}$ (henry per ohm) is a time: second.

Sums and differences

Z = A \pm B \Rightarrow \Delta Z = \Delta A + \Delta B

Worked example

Readings 2.1, 2.3, 2.2 and 2.4 s. Mean and mean absolute error?

Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 20 April Shift I · Q10Moderate

Example 1 · Units and Measurement · Units, Dimensions, and Error Analysis

A student measures time for 20 oscillations of a simple pendulum as

30\text{ }s,32\text{ }s,35\text{ }s

and 35 s . If the minimum division in the measuring clock is 1 s , then correct mean time (in second) is

Subtracting errors in a difference

Errors never cancel: the worst case is one reading high and the other low. A difference carries the SUM of the absolute errors.

Drill 3 more on means, absolute errors and kinds of error

Concept 2 of 2: Percentage Errors Through Products and Powers

For Z = AᵃBᵇ/Cᶜ, the maximum fractional error is a(ΔA/A) + b(ΔB/B) + c(ΔC/C): each error times the magnitude of its power, all added — even the ones in the denominator. A density m/l³ from a cube with 3% in l and 4% in m has 4 + 3 × 3 = 13%. A radius error of 2% gives 6% in a volume. For g from a pendulum, g ∝ l/T², so Δg/g = Δl/l + 2ΔT/T, with ΔT/T from the stopwatch's least count over the total time measured.

Definition

$Z = \dfrac{A^aB^b}{C^c}$ ⇒ $\dfrac{\Delta Z}{Z} = a\dfrac{\Delta A}{A} + b\dfrac{\Delta B}{B} + c\dfrac{\Delta C}{C}$ .
Density $\dfrac{m}{l^3}$ : 5% and 6% ⇒ 23%. Pressure $\dfrac{F}{L^2}$ : 3% and 2% ⇒ 7%. KE $\tfrac{1}{2}mv^2$ : 3% and 4% ⇒ 11%.
$\dfrac{pq^2}{r^2s^4}$ with 3, 2, 3, 1% ⇒ 17%.
Pendulum: $\dfrac{\Delta g}{g} = \dfrac{\Delta l}{l} + 2\dfrac{\Delta T}{T}$ (1 mm in 1 m, 0.1 s in 200 s ⇒ 0.2%).

Combining errors

\frac{\Delta Z}{Z} = a\frac{\Delta A}{A} + b\frac{\Delta B}{B} + c\frac{\Delta C}{C}

Worked example

Z = A²B/C³ with errors 1%, 2% and 1%. Percentage error in Z?

Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 26 April Shift I · Q39Moderate

Example 2 · Units and Measurement · Units, Dimensions, and Error Analysis

The error in the measurement of length and mass is

3\%

and

4\%

respectively. The error in the measurement of density will be

Subtracting the error of a quantity in the denominator

Dividing by C does not cancel C's error; it adds c times it. Every term in the error sum is positive.

Forgetting the power

An error in a quantity that appears cubed counts three times. A 3% error in a side gives 9% in a volume and in a density.

Drill 9 more on percentage errors through products and powers

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Means, Absolute Errors and Kinds of Error
Sums and differences
$Z = A \pm B \Rightarrow \Delta Z = \Delta A + \Delta B$
Percentage Errors Through Products and Powers
Combining errors
$\frac{\Delta Z}{Z} = a\frac{\Delta A}{A} + b\frac{\Delta B}{B} + c\frac{\Delta C}{C}$

Watch out for (3)

Subtracting errors in a difference
→ Means, Absolute Errors and Kinds of Error
Subtracting the error of a quantity in the denominator
→ Percentage Errors Through Products and Powers
Forgetting the power
→ Percentage Errors Through Products and Powers

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