Errors in Measurement

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JEE Main 2021

The acceleration due to gravity is found upto an accuracy of 4\% on a planet. The energy supplied to a simple pendulum to known mass $m$ to undertake oscillations of time period $T$ is being estimated. If time period is measured to an accuracy of 3\%, the accuracy to which $E$ is known as ..........\%.

JEE Main 2015

The period of oscillation of a simple pendulum is $T = 2 π \frac{L}{g}$ . Measured value of $L$ is $20.0 cm$ known to $1 mm$ accuracy and time for $100$ oscillations of the pendulum is found to be $90 s$ using a wrist watch of $1 s$ resolution. The accuracy in the determination of $g$ is

2%

(B)

3%

(C)

1%

(D)

5%

JEE Advanced 2016

LEVELJEE Advanced

In an experiment to determine the acceleration due to gravity $g$ , the formula used for the time period of a periodic motion is $T = 2 π \frac{7 ( R - r )}{5 g}$ . The values of $R$ and $r$ are measured to be $(60 \pm 1) mm$ and $(10 \pm 1) mm$ , respectively. In five successive measurements, the time period is found to be $0.52 s$ , $0.56 s$ , $0.57 s$ , $0.54 s$ and $0.59 s$ . The least count of the watch used for the measurement of time period is $0.01 s$ . Which of the following statement(s) is (are) true?

* Multiple Correct Options

(A)

The error in the measurement of

r

10%

(B)

The error in the measurement of

T

3.57%

(C)

The error in the measurement of

T

2%

(D)

The error in the measurement of

g

11%

JEE Advanced 2012

LEVELJEE Advanced

In the determination of Young's modulus $(Y = \frac{4 M Lg}{π l d ^{2}})$ by using Searle's method, a wire of length $L = 2 m$ and diameter $d = 0.5 mm$ is used. For a load $M = 2.5 kg$ , an extension $l = 0.25 mm$ in the length of the wire is observed. Quantities $d$ and $l$ are measured using a screw gauge and a micrometer, respectively. They have the same pitch of $0.5 mm$ . The number of divisions on their circular scale is 100. The contributions to the maximum probable error of the $Y$ measurement is

(A)

due to the errors in the measurements of

d

and

l

are the same

(B)

due to the error in the measurement of

d

is twice that due to the error in the measurement of

l

(C)

due to the error in the measurement of

l

is twice that due to the error in the measurement of

d

(D)

due to the error in the measurement of

d

is four times that due to the error in the measurement of

l

JEE Advanced 2007

LEVELJEE Advanced

A student performs an experiment to determine the Young's modulus of a wire, exactly $2 m$ long, by Searle's method. In a particular reading, the student measures the extension in the length of the wire to be $0.8 mm$ with an uncertainty of $\pm 0.05 mm$ at a load of exactly $1.0 kg$ . The student also measures the diameter of the wire to be $0.4 mm$ with an uncertainty of $\pm 0.01 mm$ . Take $g = 9.8 m/s^{2}$ (exact). The Young's modulus obtained from the reading is close to

(A)

(2.0 \pm 0.3) \times 1 0^{11} N/m^{2}

(B)

(2.0 \pm 0.2) \times 1 0^{11} N/m^{2}

(C)

(2.0 \pm 0.1) \times 1 0^{11} N/m^{2}

(D)

(2.0 \pm 0.05) \times 1 0^{11} N/m^{2}

JEE Main 2011

LEVELJEE Main

The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured with a screw gauge, whose pitch is $0.5 mm$ and there are $50$ divisions on the circular scale. The reading on the main scale is $2.5 mm$ and that on the circular scale is $20$ divisions. If the measured mass of the ball has a relative error of $2%$ , the relative percentage error in the density is

(A)

0.9%

(B)

2.4%

(C)

3.1%

(D)

4.2%

JEE Advanced 2015

LEVELJEE Advanced

The energy of a system as a function of time $t$ is given as $E (t) = A^{2} exp (- α t)$ , where $α = 0.2 s^{- 1}$ . The measurement of $A$ has an error of $1.25%$ . If the error in the measurement of time is $1.50%$ , the percentage error in the value of $E (t)$ at $t = 5 s$ is

JEE Advanced 2010

LEVELJEE Main

A student uses a simple pendulum of exactly $1 m$ length to determine $g$ , the acceleration due to gravity. He uses a stop watch with the least count of $1 s$ for this and records $40 s$ for $20$ oscillations. For this observation, which of the following statement(s) is/are true?

* Multiple Correct Options

(A)

Error

Δ T

in measuring

T

, the time period, is

0.05 s

(B)

Error

Δ T

in measuring

T

, the time period, is

1 s

(C)

Percentage error in the determination of

g

5%

(D)

Percentage error in the determination of

g

2.5%

LEVELJEE Main

Students I, II and III perform an experiment for measuring the acceleration due to gravity ( $g$ ) using a simple pendulum. They use different lengths of the pendulum and/or record time for different number of oscillations. The observations are shown in the table. Least count for length = $0.1 cm$ , Least count for time = $0.1 s$ $Student I II III Length of the pendulum (cm) 64.0 64.0 20.0 Number of oscillations (n) 844 Total time for (n) oscillations (s) 128.0 64.0 36.0 Time period (s) 16.0 16.0 9.0$ If $E_{I}$ , $E_{II}$ and $E_{III}$ are the percentage errors in $g$ , i.e. $(\frac{Δ g}{g} \times 100)$ for students I, II and III, respectively

(A)

E_{I} = 0

(B)

E_{I}

is minimum

(C)

E_{I} = E_{II}

(D)

E_{II}

is maximum

JEE Main 2004

LEVELJEE Main

A wire has a mass $(0.3 \pm 0.003) g$ , radius $(0.5 \pm 0.005) mm$ and length $(6 \pm 0.06) cm$ . The maximum percentage error in the measurement of its density is

Comprehension Passage

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation

z = x / y

. If the errors in

x, y

and

z

are

Δ x, Δ y

and

Δ z

respectively, then

z \pm Δ z = \frac{x \pm Δ x}{y \pm Δ y} = \frac{x}{y} (1 \pm \frac{Δ x}{x}) (1 \pm \frac{Δ y}{y})^{- 1}

The series expansion for

(1 \pm \frac{Δ y}{y})^{- 1}

to first power in

Δ y / y

, is

1 \mp (Δ y / y)

. The relative errors in independent variables are always added. So, the error in

z

will be

Δ z = z (\frac{Δ x}{x} + \frac{Δ y}{y})

The above derivation makes the assumption that

Δ x / x ≪ 1

Δ y / y ≪ 1

. Therefore, the higher powers of these quantities are neglected.

Question 1:

Consider the ratio $r = \frac{( 1 - a )}{( 1 + a )}$ to be determined by measuring a dimensionless quantity $a$ . If the error in the measurement of $a$ is $Δ a$ ( $Δ a / a ≪ 1$ ), then what is the error $Δ r$ in determining $r$ ?

(A)

\frac{Δ a}{( 1 + a ) ^{2}}

(B)

\frac{- 2Δ a}{( 1 + a ) ^{2}}

(C)

\frac{2Δ a}{( 1 - a ) ^{2}}

(D)

\frac{2 a Δ a}{( 1 - a ^{2} )}

Question 2:

In an experiment, the initial number of radioactive nuclei is 3000. It is found that $1000 \pm 40$ nuclei decayed in the first 1.0 s. For $∣ x ∣ ≪ 1$ , $ln (1 + x) = x$ up to first power in $x$ . The error $Δ λ$ , in the determination of the decay constant $λ$ in $s^{- 1}$ , is

Using the expression $2 d sin θ = λ$ , one calculates the values of $d$ by measuring the corresponding angles $θ$ in the range $0^{\circ}$ to $9 0^{\circ}$ . The wavelength $λ$ is exactly known and the error in $θ$ is constant for all values of $θ$ . As $θ$ increases from $0^{\circ}$

(A)

the absolute error in

d

remains constant

(B)

the absolute error in

d

increases

(C)

the fractional error in

d

remains constant

(D)

the fractional error in

d

decreases

JEE Advanced 2024

LEVELJEE Main

The dimensions of a cone are measured using a scale with a least count of $2 mm$ . The diameter of the base and the height are both measured to be $20.0 cm$ . The maximum percentage error in the determination of the volume is -

JEE Advanced 2023

LEVELJEE Main

In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $10 \pm 0.1 cm$ and the distance of its real image from the lens is $20 \pm 0.2 cm$ . The error in the determination of focal length of the lens is $n %$ . The value of $n$ is _______.

JEE Advanced 2025

LEVELJEE Advanced

A single slit diffraction experiment is performed to determine the slit width using the equation, $\frac{b d}{D} = mλ$ , where $b$ is the slit width, $D$ the shortest distance between the slit and the screen, $d$ the distance between the $m^{th}$ diffraction maximum and the central maximum, and $λ$ is the wavelength. $D$ and $d$ are measured with scales of least count of $1 cm$ and $1 mm$ , respectively. The values of $λ$ and $m$ are known precisely to be $600 nm$ and $3$ , respectively. The absolute error (in $μ m$ ) in the value of $b$ estimated using the diffraction maximum that occurs for $m = 3$ with $d = 5 mm$ and $D = 1 m$ is ___

JEE Advanced 2020

LEVELJEE Advanced

Two capacitors with capacitance values $C_{1} = 2000 \pm 10 pF$ and $C_{2} = 3000 \pm 15 pF$ are connected in series. The voltage applied across this combination is $V = 5.00 \pm 0.02 V$ . The percentage error in the calculation of the energy stored in this combination of capacitors is _______.

JEE Advanced 2019

LEVELJEE Advanced

An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at 135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens is________. [JEE(Advanced) 2019]

JEE Main 2018

LEVELJEE Main

The density of a material in the shape of a cube is determined by measuring three sides of the cube and its mass. If the relative errors in measuring the mass and length are respectively $1.5%$ and $1%$ , the maximum error in determining the density is

(A)

2.5%

(B)

3.5%

(C)

4.5%

(D)

6%

JEE Main 2021

LEVELJEE Main

In order to determine the Young's modulus of a wire of radius $0.2 cm$ (measured using a scale of least count $= 0.001 cm$ ) and length $1 m$ (measured using a scale of least count $= 1 mm$ ), a weight of mass $1 kg$ (measured using a scale of least count $= 1 g$ ) was hanged to get the elongation of $0.5 cm$ (measured using a scale of least count $0.001 cm$ ). What will be the fractional error in the value of Young's modulus determined by this experiment?

(A)

0.14%

(B)

0.9%

(C)

9%

(D)

1.4%

LEVELBoard

Resistance of a given wire is obtained by measuring the current flowing in it and the voltage difference applied across it. If the percentage errors in the measurement of the current and the voltage difference are $3%$ each, then error in the value of resistance of the wire is

(A)

6%

(B)

zero

(C)

1%

(D)

3%

JEE Main 2021

LEVELJEE Main

The period of oscillation of a simple pendulum is $T = 2 π \frac{L}{g}$ . Measured value of $L$ is $1.0 m$ from metre scale having a minimum division of $1 mm$ and time of one complete oscillation is $1.95 s$ measured from stopwatch of $0.01 s$ resolution. The percentage error in the determination of $g$ will be

(A)

1.13%

(B)

1.03%

(C)

1.33%

(D)

1.30%

JEE Main 2021

LEVELJEE Main

Two resistors $R_{1} = (4 \pm 0.8) Ω$ and $R_{2} = (4 \pm 0.4) Ω$ are connected in parallel. The equivalent resistance of their parallel combination will be

(A)

(4 \pm 0.4) Ω

(B)

(2 \pm 0.4) Ω

(C)

(2 \pm 0.3) Ω

(D)

(4 \pm 0.3) Ω

JEE Main 2020

LEVELJEE Main

A physical quantity $z$ depends on four observables $a, b, c$ and $d$ , as $z = \frac{a ^{2} b ^{2/3}}{c d ^{3}}$ . The percentages of error in the measurement of $a, b, c$ and $d$ are $2%$ , $1.5%$ , $4%$ and $2.5%$ respectively. The percentage of error in $z$ is

(A)

13.5%

(B)

16.5%

(C)

14.5%

(D)

12.25%

JEE Main 2016

LEVELJEE Main

A student measures the time period of 100 oscillations of a simple pendulum four times. The data set is $90 s$ , $91 s$ , $92 s$ and $95 s$ . If the minimum division in the measuring clock is $1 s$ , then the reported mean time should be

(A)

(92 \pm 2) s

(B)

(92 \pm 5) s

(C)

(92 \pm 1.8) s

(D)

(92 \pm 3) s

JEE Main 2019

LEVELJEE Main

In the density measurement of a cube, the mass and edge length are measured as $(10.00 \pm 0.10)$ kg and $(0.10 \pm 0.01)$ m, respectively. The error in the measurement of density is

(A)

0.01 kg/m^{3}

(B)

0.10 kg/m^{3}

(C)

0.07 kg/m^{3}

(D)

0.31 kg/m^{3}

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A wire has a mass (0.3±0.003) g, radius (0.5±0.005) mm and length (6±0.06) cm. The maximum percentage error in the measurement of its density is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two resistors R1​=(4±0.8) Ω and R2​=(4±0.4) Ω are connected in parallel. The equivalent resistance of their parallel combination will be

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In the density measurement of a cube, the mass and edge length are measured as (10.00±0.10) kg and (0.10±0.01) m, respectively. The error in the measurement of density is

A wire has a mass $(0.3 \pm 0.003) g$ , radius $(0.5 \pm 0.005) mm$ and length $(6 \pm 0.06) cm$ . The maximum percentage error in the measurement of its density is

Two resistors $R_{1} = (4 \pm 0.8) Ω$ and $R_{2} = (4 \pm 0.4) Ω$ are connected in parallel. The equivalent resistance of their parallel combination will be

In the density measurement of a cube, the mass and edge length are measured as $(10.00 \pm 0.10)$ kg and $(0.10 \pm 0.01)$ m, respectively. The error in the measurement of density is