Can a physical quantity have units but no dimensions?

Yes. Quantities like plane angle (measured in radians) and solid angle (measured in steradians) possess SI units but are dimensionless with dimensional formula $[M^0 L^0 T^0]$. However, the converse is impossible: a dimensionless quantity can never have units.

Why do we always add percentage errors even if terms are in the denominator?

Errors represent maximum uncertainty or worst-case tolerance limits in measurement. Since uncertainties compound during division just as they do during multiplication, subtracting errors would unrealistically underestimate the total potential deviation.

How should I handle zero error corrections in screw gauge and vernier caliper questions?

Always remember the golden rule: $\text{True Reading} = \text{Observed Reading} - (\text{Zero Error with Sign})$. A positive zero error is subtracted, whereas a negative zero error effectively gets added due to the minus sign.

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Solved High-Yield Sample Questions

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Question #1NEET 2024

A physical quantity $P$ is related to four observables $a$ , $b$ , $c$ , and $d$ as $P = \frac{a^3 b^2}{\sqrt{c} d}$ . The percentage errors of measurement in $a$ , $b$ , $c$ , and $d$ are $1\%$ , $2\%$ , $4\%$ , and $7\%$ respectively. Determine the maximum percentage error in $P$ .

Solution Locked

Step-by-Step Solution & Concept

Step 1

Apply power rule of error propagation: $\frac{\Delta P}{P} = 3\frac{\Delta a}{a} + 2\frac{\Delta b}{b} + \frac{1}{2}\frac{\Delta c}{c} + \frac{\Delta d}{d}$ .

Step 2

Substitute given percentage errors: $\% \Delta P = 3(1\%) + 2(2\%) + \frac{1}{2}(4\%) + 1(7\%)$ .

Step 3

Sum the individual percentage contributions: $\% \Delta P = 3\% + 4\% + 2\% + 7\% = 16\%$ .

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Question #2JEE Main 2023

In a Vernier Calipers, $10$ divisions of the Vernier scale coincide with $9$ divisions of the main scale. If $1\text{ MSD} = 1\text{ mm}$ , and while measuring diameter of a sphere, main scale reading is $2.8\text{ cm}$ and $7^{\text{th}}$ vernier division coincides, find measured diameter given zero error is $-0.03\text{ cm}$ .

Solution Locked

Step-by-Step Solution & Concept

Step 1

Calculate Least Count ( $\text{LC}$ ): $\text{LC} = 1\text{ MSD} - 1\text{ VSD} = 1\text{ mm} - 0.9\text{ mm} = 0.1\text{ mm} = 0.01\text{ cm}$ .

Step 2

Calculate Observed Reading: $\text{OR} = \text{MSR} + (\text{VSR} \times \text{LC}) = 2.8\text{ cm} + (7 \times 0.01\text{ cm}) = 2.87\text{ cm}$ .

Step 3

Apply zero correction: $\text{True Reading} = \text{OR} - (\text{Zero Error}) = 2.87\text{ cm} - (-0.03\text{ cm}) = 2.90\text{ cm}$ .

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Mastering Units and Measurements: Formulas, Traps & Exam Strategy | ExamiFly

Chapter Overview

Core Subtopics & Syllabus Breakdown

Key Formulas & High-Yield Equations

Principle of Homogeneity of Dimensions

Relative and Percentage Error

Error Propagation for Power Relations

Least Count of Vernier Caliper

Least Count of Screw Gauge

Common Mistakes & Traps to Avoid

Past Year Question (PYQ) Trends

Solved High-Yield Sample Questions

Frequently Asked Questions

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