Error of Measurement

Q 11 :

A quantity Q is formulated as $X^{- 2} Y^{+ \frac{3}{2}} Z^{- \frac{2}{5}} \cdot X, Y$ , and Z are independent parameters which have fractional errors of 0.1, 0.2 and 0.5, respectively in measurement. The maximum fractional error of Q is [2025]

0.1
0.8
0.7
0.6

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3

4

(3)

Fractional error $= 2 \frac{△ X}{X} + \frac{3}{2} \frac{△ Y}{Y} + \frac{2}{5} \frac{△ Z}{Z}$

$= 2 (0.1) + \frac{3}{2} (0.2) + \frac{2}{5} (0.5)$

= 0.2 + 0.3 + 0.2 = 0.7

Q 12 :

A physical quantity Q is related to four observables a, b, c, d as follows:

$Q = \frac{a b^{4}}{c d} where, a = (60 \pm 3) P a; b = (20 \pm 0.1) m$ ,

$c = (40 \pm 0.2) {Nsm}^{- 2} and d = (50 \pm 0.1) m$ ,

then the percentage error in Q is $\frac{x}{1000}$ , where x = __________. [2025]

0%

(77)

$Q = \frac{a b^{4}}{c d}$

$\Rightarrow \frac{△ Q}{Q} = [\frac{△ a}{a} + 4 \frac{△ b}{b} + \frac{△ c}{c} + \frac{△ d}{d}]$

$\Rightarrow \frac{x}{1000} = [\frac{3}{60} + 4 (\frac{0.1}{20}) + (\frac{0.2}{40}) + \frac{0.1}{50}]$

$\Rightarrow x = 77$

Q 13 :

A physical quantity C is related to four other quantities p, q, r and s as follows

$C = \frac{p q^{2}}{r^{3} \sqrt{s}}$

The percentage errors in the measurement of p, q, r and s are 1%, 2%, 3% and 2% respectively. The percentage error in the measurement of C will be __________ %. [2025]

0%

(15)

$C = p^{1} q^{2} r^{- 3} s^{- 1 / 2}$

${(\frac{d C}{C})}_{\max} = \frac{d P}{P} + \frac{2 d q}{q} + \frac{3 d r}{r} + \frac{1}{2} \frac{d s}{s}$

$= (1 + 2 \times 2 + 3 \times 3 + \frac{1}{2} \times 2) % = 15 %$

Q 14 :

Two resistances are given as $R_{1} = (10 \pm 0.5) Ω$ and $R_{2} = (15 \pm 0.5) Ω$ . The percentage error in the measurement of equivalent resistance when they are connected in parallel is [2023]

6.33
5.33
2.33
4.33

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4

(4)

$\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$

Differentiating both sides, we get

$\frac{△ R}{R^{2}} = \frac{△ R_{1}}{R_{1}^{2}} + \frac{△ R_{2}}{R_{2}^{2}} [R = \frac{R_{1} R_{2}}{R_{1} + R_{2}} = \frac{10 \times 15}{10 + 15} = 6]$

$\Rightarrow \frac{△ R}{R} = (\frac{△ R_{1}}{R_{1}^{2}} + \frac{△ R_{2}}{R_{2}^{2}}) R$

$\Rightarrow (\frac{0.5}{100} + \frac{0.5}{225}) 6 = (\frac{6 \times 0.5}{25}) (\frac{1}{4} + \frac{1}{9}) = \frac{13}{100}$

$\frac{△ R}{R} \times 100 = \frac{13}{3} = 4.33 %$

Q 15 :

A 2 meter long scale with least count of 0.2 cm is used to measure the locations of objects on an optical bench. While measuring the focal length of a convex lens, the object pin and the convex lens are placed at 80 cm mark and 1 m mark, respectively. The image of the object pin on the other side of lens coincides with image pin that is kept at 180 cm mark. The % error in the estimation of focal length is [2023]

1.02
0.85
1.70
0.51

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4

(3)

Least count of the scale is 0.2 cm

u = (100 $\pm$ 0.2) – (80 $\pm$ 0.2) = (20 $\pm$ 0.4) cm

v = (180 $\pm$ 0.2) – (100 $\pm$ 0.2) = (80 $\pm$ 0.4) cm

From lens formula,

$\frac{1}{v} - \frac{1}{u} = \frac{1}{f} \Rightarrow \frac{1}{f} = \frac{1}{80} - \frac{1}{- 20} \Rightarrow f = 16 cm$

Also, $\frac{△ v}{v^{2}} + \frac{△ u}{u^{2}} = \frac{△ f}{f^{2}}$

$\Rightarrow \frac{△ f}{f} \times 100 = (\frac{△ v}{v^{2}} + \frac{△ u}{u^{2}}) \times f \times 100$

$\Rightarrow % f = (\frac{0.4}{400} + \frac{0.4}{6400}) \times 16 \times 100 = 1.70$

Q 16 :

A cylindrical wire of mass (0.4 $\pm$ 0.01) g has length (8 $\pm$ 0.04) cm and radius (6 $\pm$ 0.03) cm. The maximum error in its density will be [2023]

1%
4%
5%
3.5%

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3

4

(2)

$ρ = \frac{m}{π r^{2} l} \Rightarrow$ $| \frac{d ρ}{ρ} |_{\max}$ $=$ $| \frac{d m}{m} |$ $+ 2$ $| \frac{d r}{r} |$ $+$ $| \frac{d l}{l} |$

$= \frac{0.01}{0.4} + \frac{2 (0.03)}{6} + \frac{0.04}{8}$

$\Rightarrow % error in density = (\frac{d ρ}{ρ}) \times 100 %$

$= (2.5 + 1 + 0.5) % = 4 %$

Q 17 :

A physical quantity P is given as $P = \frac{a^{2} b^{3}}{c \sqrt{d}}$ . The percentage error in the measurement of a, b, c and d are 1%, 2%, 3% and 4% respectively. The percentage error in the measurement of quantity P will be [2023]

13%
14%
12%
16%

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3

4

(1)

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

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

Q 18 :

A body of mass (5 $\pm$ 0.5) kg is moving with a velocity of (20 $\pm$ 0.4) m/s. Its kinetic energy will be [2023]

(1000 $\pm$ 140) J
(1000 $\pm$ 0.14) J
(500 $\pm$ 0.14) J
(500 $\pm$ 140) J

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(1)

$k = \frac{1}{2} m v^{2}$

$k = \frac{1}{2} \times 5 \times 400 = 5 \times 200 = 1000 J$

$\frac{△ k}{k} = \frac{△ m}{m} + \frac{2 △ v}{v} = \frac{0.5}{5} + \frac{2 \times 0.4}{20}$

$△ k = 1000 (\frac{1}{10} + \frac{4}{100}) = 1000 (\frac{10 + 4}{100}) = 140 J$

Topic Question Set