Accuracy Precision of Instruments and Errors in Measurement

Q 1 :
A student measured the diameter of a small steel ball using a screw gauge of least count 0.001 cm. The main scale reading is 5 mm and zero of circular scale division coincides with 25 divisions above the reference level.

If screw gauge has a zero error of -0.004 cm, the correct diameter of the ball is [2018]

0.529 cm
0.521 cm
0.053 cm
0.525 cm

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(1)

Measured diameter of the ball = Reading of screw gauge

= MSR + VSR x LC + zero error

= 0.5 cm + 25 x 0.001 cm + 0.004 cm = 0.529 cm

Q 2 :
In an experiment, the percentage of error occurred in the measurement of physical quantities A, B, C and D are 1%, 2%, 3% and 4%, respectively. The maximum percentage of error in the measurement X, where $X = \frac{A^{2} B^{1 / 2}}{C^{1 / 3} D^{3}}$ , will be [2019]

10%
$(\frac{3}{13}) %$
16%
-10%

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(3)

$X = \frac{A^{2} B^{1 / 2}}{C^{1 / 3} D^{3}}$

Maximum percentage error in X

$(\frac{d X}{X}) \times 100 = (2 \frac{d A}{A} + \frac{1}{2} \frac{d B}{B} + \frac{1}{3} \frac{d C}{C} + 3 \frac{d D}{D}) \times 100$

$= 2 \times 1 + \frac{1}{2} \times 2 + \frac{1}{3} \times 3 + 3 \times 4 = 16 %$

Q 3 :
A screw gauge has least count of 0.01 mm and there are 50 divisions in its circular scale. The pitch of the screw gauge is [2020]

1.0 mm
0.01 mm
0.25 mm
0.5 mm

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(4)

$L . C . = \frac{P i t c h}{N u m b e r o f d i v i s i o n s o n c i r c u l a r s c a l e}$

$0.01 = \frac{P i t c h}{50}$

$P i t c h = 0.5 m m$

Q 4 :
A screw gauge gives the following readings when used to measure the diameter of a wire

Main scale reading : 0 mm

Circular scale reading : 52 divisions

Given that 1 mm on main scale corresponds to 100 divisions on the circular scale. The diameter of the wire from the above data is (2021)

0.52 cm
0.026 cm
0.26 cm
0.052 cm

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(4)

Given the pitch of the screw gauge, P = 1 mm

Number of circular division, n = 100

Thus least count $L C = \frac{p}{n} = \frac{1}{100} = 0.01 m m = 0.001 c m$

So, diameter of the wire $= M S R + (C S R \times L C)$

=0 + (52 x 0.001 cm) = 0.052 cm

Q 5 :
A metal wire has mass $(0.4 \pm 0.002)$ g, radius $(0.3 \pm 0.001)$ mm and length $(5 \pm 0.02)$ cm. The maximum possible percentage error in the measurement of density will nearly be [2023]

1.6%
1.4%
1.2%
1.3%

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(1)

Volume of the wire, $V = π r^{2} L$

Density of the wire $ρ = \frac{m}{V} = \frac{m}{π r^{2} L}$

For percentage error,

$\frac{Δ ρ}{ρ} \times 100 = \frac{Δ m}{m} \times 100 + 2 \times \frac{Δ r}{r} \times 100 + \frac{Δ L}{L} \times 100$

$∴$ $\frac{Δ ρ}{ρ} \times 100 = \frac{0.002}{0.4} \times 100 + 2 \times \frac{0.001 \times 100}{0.3} + \frac{0.02}{5} \times 100$

$= 0.5 + 0.67 + 0.4 = 1.57 \approx 1.6 %$

Q 6 :
The errors in the measurement which arise due to unpredictable fluctuations in temperature and voltage supply are [2023]

Least count errors
Random errors
Instrumental errors
Personal errors

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(2)

Random errors are the errors which occur due to random and unpredictable fluctuations in temperature and voltage supply.

Q 7 :
The main scale of a vernier callipers has n divisions/cm. n divisions of the vernier scale coincide with (n-1) divisions of main scale. The least count of the vernier callipers is [2019]

$\frac{1}{(n + 1) (n - 1)} c m$
$\frac{1}{n} c m$
$\frac{1}{n^{2}} c m$
$\frac{1}{n (n + 1)} c m$

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(3)

If n divisions of the vernier scale coincide with (n−1) divisions of the main scale:

Therefore, $n V S D = (n - 1) M S D \Rightarrow 1 V S D = \frac{(n - 1)}{n} M S D$

Least count= $1 M S D - 1 V S D = 1 M S D - \frac{(n - 1)}{n} M S D = 1 M S D - 1 M S D + \frac{1}{n} M S D = \frac{1}{n} M S D$
= $\frac{1}{n} \times \frac{1}{n} = \frac{1}{n^{2}} c m$ $⋮ [1 M S D = \frac{1}{n} c m]$

Q 8 :
A physical quantity P is related to four observations a, b, c and d as follows: $P = \frac{a^{3} b^{2}}{c \sqrt{d}}$

The percentage errors of measurement in a, b, c and d are 1%, 3%, 2% and 4% respectively. The percentage error in the quantity P is: [2025]

13%
15%
10%
2%

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(1)

Given, $P = \frac{a^{3} b^{2}}{c \sqrt{d}}$

Percentage error in quantity P,

$\frac{Δ P}{P} \times 100 = (\frac{3 Δ a}{a} + \frac{2 Δ b}{b} + \frac{Δ c}{c} + \frac{1}{2} \frac{Δ d}{d}) \times 100$

$= 3 (1) + 2 (3) + 2 + \frac{1}{2} \times 4 = 3 + 6 + 2 + 2 = 13 %$

Topic Question Set

Q 2 :
In an experiment, the percentage of error occurred in the measurement of physical quantities A, B, C and D are 1%, 2%, 3% and 4%, respectively. The maximum percentage of error in the measurement X, where $X = \frac{A^{2} B^{1 / 2}}{C^{1 / 3} D^{3}}$ , will be [2019]

Q 3 :
A screw gauge has least count of 0.01 mm and there are 50 divisions in its circular scale. The pitch of the screw gauge is [2020]

Q 4 :
A screw gauge gives the following readings when used to measure the diameter of a wire

Main scale reading : 0 mm

Circular scale reading : 52 divisions

Given that 1 mm on main scale corresponds to 100 divisions on the circular scale. The diameter of the wire from the above data is (2021)

Q 5 :
A metal wire has mass $(0.4 \pm 0.002)$ g, radius $(0.3 \pm 0.001)$ mm and length $(5 \pm 0.02)$ cm. The maximum possible percentage error in the measurement of density will nearly be [2023]

Q 6 :
The errors in the measurement which arise due to unpredictable fluctuations in temperature and voltage supply are [2023]

Q 7 :
The main scale of a vernier callipers has n divisions/cm. n divisions of the vernier scale coincide with (n-1) divisions of main scale. The least count of the vernier callipers is [2019]

Q 8 :
A physical quantity P is related to four observations a, b, c and d as follows: $P = \frac{a^{3} b^{2}}{c \sqrt{d}}$

The percentage errors of measurement in a, b, c and d are 1%, 3%, 2% and 4% respectively. The percentage error in the quantity P is: [2025]

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Topic Question Set

Q 2 : In an experiment, the percentage of error occurred in the measurement of physical quantities A, B, C and D are 1%, 2%, 3% and 4%, respectively. The maximum percentage of error in the measurement X, where X=A2B1/2C1/3D3 , will be [2019]

Q 3 : A screw gauge has least count of 0.01 mm and there are 50 divisions in its circular scale. The pitch of the screw gauge is [2020]

Q 4 : A screw gauge gives the following readings when used to measure the diameter of a wire Main scale reading : 0 mm Circular scale reading : 52 divisions Given that 1 mm on main scale corresponds to 100 divisions on the circular scale. The diameter of the wire from the above data is (2021)

Q 5 : A metal wire has mass (0.4±0.002) g, radius (0.3±0.001) mm and length (5±0.02) cm. The maximum possible percentage error in the measurement of density will nearly be [2023]

Q 6 : The errors in the measurement which arise due to unpredictable fluctuations in temperature and voltage supply are [2023]

Q 7 : The main scale of a vernier callipers has n divisions/cm. n divisions of the vernier scale coincide with (n-1) divisions of main scale. The least count of the vernier callipers is [2019]

Q 8 : A physical quantity P is related to four observations a, b, c and d as follows: P=a3b2cd The percentage errors of measurement in a, b, c and d are 1%, 3%, 2% and 4% respectively. The percentage error in the quantity P is: [2025]

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Q 2 :
In an experiment, the percentage of error occurred in the measurement of physical quantities A, B, C and D are 1%, 2%, 3% and 4%, respectively. The maximum percentage of error in the measurement X, where $X = \frac{A^{2} B^{1 / 2}}{C^{1 / 3} D^{3}}$ , will be [2019]

Q 3 :
A screw gauge has least count of 0.01 mm and there are 50 divisions in its circular scale. The pitch of the screw gauge is [2020]

Q 4 :
A screw gauge gives the following readings when used to measure the diameter of a wire

Main scale reading : 0 mm

Circular scale reading : 52 divisions

Given that 1 mm on main scale corresponds to 100 divisions on the circular scale. The diameter of the wire from the above data is (2021)

Q 5 :
A metal wire has mass $(0.4 \pm 0.002)$ g, radius $(0.3 \pm 0.001)$ mm and length $(5 \pm 0.02)$ cm. The maximum possible percentage error in the measurement of density will nearly be [2023]

Q 6 :
The errors in the measurement which arise due to unpredictable fluctuations in temperature and voltage supply are [2023]

Q 7 :
The main scale of a vernier callipers has n divisions/cm. n divisions of the vernier scale coincide with (n-1) divisions of main scale. The least count of the vernier callipers is [2019]

Q 8 :
A physical quantity P is related to four observations a, b, c and d as follows: $P = \frac{a^{3} b^{2}}{c \sqrt{d}}$

The percentage errors of measurement in a, b, c and d are 1%, 3%, 2% and 4% respectively. The percentage error in the quantity P is: [2025]