Measuring Instruments, Errors and the Best-Fit Line: Quick Revision

A brief note on Measuring Instruments, Errors and the Best-Fit Line. Read it before a practical examination. Try each question at the end before you open its answer.

The instruments

• Vernier calipers: $\text{LC} = \dfrac{1\ \text{MSD}}{N}$ and reading $= \text{MSR} + n \times \text{LC}$. The usual least count is $0.1$ mm.
• Screw gauge: $\text{LC} = \dfrac{\text{pitch}}{N}$ and reading $= \text{PSR} + \text{CSR} \times \text{LC}$. The usual least count is $0.01$ mm.
• Zero error: corrected reading $=$ observed reading $-$ zero error. Positive: $+k \times \text{LC}$. Negative: $-(N - k) \times \text{LC}$.
• Good habits: take many readings, in two directions, and use the ratchet to close the screw.

Digits and errors

• Significant figures: leading zeros do not count. In a sum keep the fewest decimal places. In a product keep the fewest significant figures.
• Errors of a set of readings: the mean, then $\Delta a_i = |a_{mean} - a_i|$, then the mean of these. The relative error is $\Delta a / a$.
• Never below the least count: a result is not more exact than the instrument.
• Sum or difference: the absolute errors add.
• Product, quotient or power: $\dfrac{\Delta Z}{Z} = p\dfrac{\Delta A}{A} + q\dfrac{\Delta B}{B}$.

The pendulum and the line

• The pendulum: $T = 2\pi\sqrt{L/g}$, so $g = \dfrac{4\pi^2 L}{T^2}$. $L$ goes to the centre of the bob. Time 20 oscillations or more.
• The graph: $T^2$ against $L$ is a straight line. The slope is $\dfrac{4\pi^2}{g}$. A missing bob radius shifts the line, not its slope.
• The metre scale: its centre of gravity is at $50.0$ cm. Moments about the knife edge balance.
• A best line: many points, one thin line, and a slope from two far-apart points on the line.

A worked example

A screw gauge has a pitch of $1$ mm and $100$ divisions. The closed reading is $4$. A wire gives a pitch scale reading of $1$ mm and a circular reading of $38$. Find the diameter.

1. The least count. $\dfrac{1\ \text{mm}}{100} = 0.01$ mm.
2. The zero error. $4$ is small, so the error is $+4 \times 0.01\ \text{mm} = +0.04$ mm.
3. The diameter. $1\ \text{mm} + 38 \times 0.01\ \text{mm}$ $- 0.04\ \text{mm}$ $= 1.34$ mm.

Practise with new numbers

Check yourself

1. The time for 20 oscillations is $44.0$ s with a stopwatch of least count $0.1$ s. What is the percentage error in $T$?

2. A length is $x = 5.0 \pm 0.1$ cm and a second is $y = 3.0 \pm 0.1$ cm. What is $x - y$ with its error?

3. The graph of $T^2$ against $L$ for a pendulum has a slope of $4.00\ \text{s}^2/\text{m}$. What is g?

Sources

• NCERT, Laboratory Manual: Physics, Class XI.
• OpenStax University Physics Volume 1, Chapter 1 (significant figures) and Chapter 15 (the simple pendulum).