Error of Measurement

Q 1 :
In an experiment to measure focal length $(f)$ of convex lens, the least count of the measuring scales for the position of object ( $u$ ) and for the position of image ( $v$ ) are $∆ u$ and $∆ v,$ respectively. The error in the measurement of the focal length of the convex lens will be [2024]

$2 f [\frac{∆ u}{u} + \frac{∆ v}{v}]$
$f [\frac{∆ u}{u} + \frac{∆ v}{v}]$
$\frac{∆ u}{u} + \frac{∆ v}{v}$
$f^{2} [\frac{∆ u}{u^{2}} + \frac{∆ v}{v^{2}}]$

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(D) $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$

$\frac{- 1}{v^{2}} d v + \frac{1}{u^{2}} d u = \frac{- 1}{f^{2}} d f \Rightarrow \frac{d f}{f^{2}} = \frac{d v}{v^{2}} + \frac{d u}{u^{2}}$

$d f = f^{2} [\frac{d v}{v^{2}} + \frac{d u}{u^{2}}] \Rightarrow ∆ f = f^{2} (\frac{∆ v}{v^{2}} + \frac{∆ u}{u^{2}})$

Q 2 :
To find the spring constant ( $k$ ) of a spring experimentally, a student commits 2% positive error in the measurement of time and 1% negative error in measurement of mass. The percentage error in determining value of $k$ is: [2024]

3%
4%
5%
1%

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(A) $T = 2 π \sqrt{\frac{m}{k}} \Rightarrow k = 4 π^{2} \cdot \frac{m}{T^{2}}$

$\frac{∆ k}{k} = \frac{∆ m}{m} + 2 \cdot \frac{∆ T}{T} \Rightarrow \frac{∆ k}{k} % = \frac{∆ m}{m} % + \frac{2 ∆ T}{T} %$

(We always take maximum error)

$\frac{∆ k}{k} % = (- 1) % + 2 (2) % = | - 3 % | = 3 %$

Q 3 :
Young's modulus is determined by the equation given by $Y = 49000 \frac{m}{L} \frac{d y n e}{c m^{2}}$ where M is the mass and L is the extension of wire used in the experiment. Now error in Young's modules ( $Y$ ) is estimated by taking data from $M - L$ plot in graph paper. The smallest scale divisions are 5g and 0.02 cm along load axis and extension axis respectively. If the value of $M$ and are 500 g and 2 cm respectively then percentage error of $Y$ is [2024]

0.2%
0.02%
2%
0.5%

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(C) $\frac{∆ Y}{Y} = \frac{∆ m}{m} + \frac{∆ l}{l}$

$= \frac{5}{500} + \frac{0.02}{2} = 0.01 + 0.01 \frac{∆ Y}{Y}$

$% \frac{∆ Y}{Y} = 2 %$

Q 4 :
The resistance $R = V / I$ where $V = (200 \pm 5) V$ and $I = (20 \pm 0.2) A .$ The percentage error in the measurement of R is: [2024]

3.5%
7%
3%
5.5%

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(A) We have, $R = V / I$

According to error analysis, maximum fractional error.

$\frac{d R}{R} = \frac{d V}{V} + \frac{d I}{I}$

$\frac{d R}{R} = \frac{5}{200} + \frac{0.2}{20} = \frac{7}{200}$

Hence, % error $\frac{d R}{R} \times 100 = \frac{7}{200} \times 100 = 3.5 %$

Q 5 :
A physical quantity $Q$ is found to depend on quantities $a, b, c$ by the relation $Q = \frac{a^{4} b^{3}}{c^{2}} .$ The percentage error in $a, b$ and $c$ are 3%, 4% and 5% respectively. Then, the percentage error in $Q$ is [2024]

66%
14%
34%
43%

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4

(C) $\frac{∆ Q}{Q} = 4 \frac{∆ a}{a} + 3 \frac{∆ b}{b} + 2 \frac{∆ c}{c}$

$Q = 4 \times 3 % + 3 \times 4 % + 2 \times 5 %$

$= 12 % + 12 % + 10 % = 34 %$

Q 6 :
If the percentage errors in measuring the length and the diameter of a wire are 0.1% each. The percentage error in measuring its resistance will be [2024]

0.2%
0.144%
0.3%
0.1%

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(C) $R = \frac{ρ L}{π \frac{d^{2}}{4}}$

$\frac{∆ R}{R} = \frac{∆ L}{L} + \frac{2 ∆ d}{d}$

$\frac{∆ L}{L} = 0.1 %$ and $\frac{∆ d}{d} = 0.1 %$

$\frac{∆ R}{R} = 0.3 %$

Q 7 :
The measured value of the length of a simple pendulum is 20 cm with 2 mm accuracy. The time for 50 oscillations was measured to be 40 seconds with 1 second resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is N%. The value of N is: [2024]

4
8
6
5

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(C) $T = 2 π \sqrt{\frac{l}{g}}$

$g = \frac{4 π^{2} l}{T^{2}}$

$\frac{∆ g}{g} = \frac{∆ l}{l} + \frac{2 ∆ T}{T} = \frac{0.2}{20} + 2 (\frac{1}{40}) = \frac{1.2}{20}$

Percentage change = $\frac{1.2}{20} \times 100 = 6 %$

Q 8 :
The radius ( $r$ ), length ( $l$ ) and resistance ( $R$ ) of a metal wire was measured in the laboratory as [2024]

$r = (0.35 \pm 0.05) c m$

$R = (100 \pm 10) o h m$

$l = (15 \pm 0.2) c m$

The percentage error in resistivity of the material of the wire is

25.6%
39.9%
37.3%
35.6%

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(B) $\frac{∆ ρ}{ρ} = \frac{∆ R}{R} + 2 \frac{∆ r}{r} + \frac{∆ l}{l}$

$= \frac{10}{100} + 2 \times \frac{0.05}{0.35} + \frac{0.2}{15}$

$= \frac{1}{10} + \frac{2}{7} + \frac{1}{75}$

$\frac{∆ ρ}{ρ} = 39.9 %$

Topic Question Set

Q 2 :
To find the spring constant ( $k$ ) of a spring experimentally, a student commits 2% positive error in the measurement of time and 1% negative error in measurement of mass. The percentage error in determining value of $k$ is: [2024]

Q 4 :
The resistance $R = V / I$ where $V = (200 \pm 5) V$ and $I = (20 \pm 0.2) A .$ The percentage error in the measurement of R is: [2024]

Q 5 :
A physical quantity $Q$ is found to depend on quantities $a, b, c$ by the relation $Q = \frac{a^{4} b^{3}}{c^{2}} .$ The percentage error in $a, b$ and $c$ are 3%, 4% and 5% respectively. Then, the percentage error in $Q$ is [2024]

Q 6 :
If the percentage errors in measuring the length and the diameter of a wire are 0.1% each. The percentage error in measuring its resistance will be [2024]

Q 7 :
The measured value of the length of a simple pendulum is 20 cm with 2 mm accuracy. The time for 50 oscillations was measured to be 40 seconds with 1 second resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is N%. The value of N is: [2024]

Q 8 :
The radius ( $r$ ), length ( $l$ ) and resistance ( $R$ ) of a metal wire was measured in the laboratory as [2024]

$r = (0.35 \pm 0.05) c m$

$R = (100 \pm 10) o h m$

$l = (15 \pm 0.2) c m$

The percentage error in resistivity of the material of the wire is

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Topic Question Set

Q 1 : In an experiment to measure focal length (f) of convex lens, the least count of the measuring scales for the position of object (u) and for the position of image (v) are ∆u and ∆v, respectively. The error in the measurement of the focal length of the convex lens will be [2024]

Q 2 : To find the spring constant (k) of a spring experimentally, a student commits 2% positive error in the measurement of time and 1% negative error in measurement of mass. The percentage error in determining value of k is: [2024]

Q 4 : The resistance R=V/I where V=(200±5)V and I=(20±0.2)A. The percentage error in the measurement of R is: [2024]

Q 5 : A physical quantity Q is found to depend on quantities a,b,c by the relation Q=a4b3c2. The percentage error in a,b and c are 3%, 4% and 5% respectively. Then, the percentage error in Q is [2024]

Q 6 : If the percentage errors in measuring the length and the diameter of a wire are 0.1% each. The percentage error in measuring its resistance will be [2024]

Q 7 : The measured value of the length of a simple pendulum is 20 cm with 2 mm accuracy. The time for 50 oscillations was measured to be 40 seconds with 1 second resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is N%. The value of N is: [2024]

Q 8 : The radius (r), length (l) and resistance (R) of a metal wire was measured in the laboratory as [2024] r=(0.35±0.05)cm R=(100±10)ohm l=(15±0.2)cm The percentage error in resistivity of the material of the wire is

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Q 2 :
To find the spring constant ( $k$ ) of a spring experimentally, a student commits 2% positive error in the measurement of time and 1% negative error in measurement of mass. The percentage error in determining value of $k$ is: [2024]

Q 4 :
The resistance $R = V / I$ where $V = (200 \pm 5) V$ and $I = (20 \pm 0.2) A .$ The percentage error in the measurement of R is: [2024]

Q 5 :
A physical quantity $Q$ is found to depend on quantities $a, b, c$ by the relation $Q = \frac{a^{4} b^{3}}{c^{2}} .$ The percentage error in $a, b$ and $c$ are 3%, 4% and 5% respectively. Then, the percentage error in $Q$ is [2024]

Q 6 :
If the percentage errors in measuring the length and the diameter of a wire are 0.1% each. The percentage error in measuring its resistance will be [2024]

Q 7 :
The measured value of the length of a simple pendulum is 20 cm with 2 mm accuracy. The time for 50 oscillations was measured to be 40 seconds with 1 second resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is N%. The value of N is: [2024]

Q 8 :
The radius ( $r$ ), length ( $l$ ) and resistance ( $R$ ) of a metal wire was measured in the laboratory as [2024]

$r = (0.35 \pm 0.05) c m$

$R = (100 \pm 10) o h m$

$l = (15 \pm 0.2) c m$

The percentage error in resistivity of the material of the wire is