Multiplication & Division of Errors

Q. The resistance $R=\frac{V}{I}$, where $V=100\pm 5$ volts and $I=10\pm 0.3$ amperes. Calculate the percentage error in $R$.

1. 8 %
2. 10 %
3. 14 %
4. 12 %

Q. In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency $500Hz$ is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length $50.7cm$ and $83.9cm$. Which of the following statements is (are) true?

1. The speed of sound determined from this experiment is $332m{s}^{-1}$
2. The end correction in this experiment is $0.9cm$
3. The wavelength of the sound wave is $66.4cm$
4. The resonance at $50.7cm$ corresponds to the fundamental harmonic

Q. The length of cylinder is measured with a metre rod having least count $0.1\text{cm}$. Its diameter is measured with vernier caliper having least count $0.01\text{cm}$. Given that length is $5.0\text{cm}$ and diameter is $4.0\text{cm}$. The percentage error in the calculated value of the volume will be

1. $1.5\%$
2. $2.5\%$
3. $3.5\%$
4. $4\%$

Q. The colour code of a carbon resistor is given below.

The value of resistance with tolerance is given as

1. $(4.2\text{k}\Omega \pm 5\%)$
2. $(4.2\text{k}\Omega \pm 10\%)$
3. $(4.3\text{k}\Omega \pm 10\%)$
4. $(2.4\text{k}\Omega \pm 5\%)$

Q. A thin copper wire of length $l\text{metre}$ increases in length by $2\%$ when heated through ${10}^{\circ }C.$ What is the percentage increase in area , when a square copper sheet of length $l\text{metre}$ is heated through ${10}^{\circ }C$ ?

1. $4\%$
2. $8\%$
3. $16\%$
4. None of the above

Q. In the circuit shown above, the key is pressed at time $t=0$. Which of the following statement(s) is(are) true?

1. The voltmeter displays $-5V$ as soon as the key is pressed, and displays $+5V$ after a long time
2. The voltmeter will display 0 V at time $t=ln$ 2 seconds
3. The current in the ammeter becomes $1/e$ of the initial value after $1$ second
4. The current in the ammeter becomes zero after a long time.

Q. According to Joule's law of heating, heat produced $H=I^{2}Rt,$ where $I$ is current, $R$ is resistance and $t$ is time. If the errors in the measurements of $I$, $R$ and $t$ are 3%, 4% and 6% respectively, then error in the measurement of H is

1. $\pm 19\%$
2. $\pm 17\%$
3. $\pm 16\%$
4. $\pm 25\%$

Q. Length $l$ and Acceleration due to gravity $g$ are measured with $\pm 6\%$ and $\pm 4\%$ errors respectively. The percentage error in determination of the Time period of a simple pendulum will be,

1. $\pm 5\%$
2. $\pm 4\%$
3. $\pm 7\%$
4. $\pm 2\%$

Q. In a meter bridge experiment, resistances are connected as shown in figure. The balancing length $l_1$ is $55\text{cm}$. Now an unknown resistance $x$ is connected in series with $P$ and the new balancing length is found to be $75\text{cm}$. The value of $x$ is

1. $\frac{54}{13}\Omega$
2. $\frac{20}{11}\Omega$
3. $\frac{48}{11}\Omega$
4. $\frac{11}{48}\Omega$

Q.

The density of the 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:

1. $4.5\%$

2. 6%

3. $2.5\%$

4. $3.5\%$

Q.

A sphere increases its volume at the rate of $\pi c{m}^3/s$. The rate at which its surface area increases, when the radius is $1cm$ is

1. $2\pi sqcm/s$

2. $\pi sqcm/s$

3. $\frac{3\pi }{2}sqcm/s$

4. $\frac{\pi }{2}sqcm/s$

Q. In a meter bridge, the gaps are closed by resistance $2\Omega$ and $3\Omega$. The value of shunt to be added to $3\Omega$ resistor to shift the balancing point by $22.5\text{cm}$ is

1. $5\Omega$
2. $2.5\Omega$
3. $2\Omega$
4. $1\Omega$

Q. In a meter bridge experiment, the value of unknown resistance is 2$\Omega$. To get the balancing point at $40$ $\text{cm}$ distance from the same end, then what will be the resistance in the resistance box?

1. $3\Omega$
2. $6\Omega$
3. $8\Omega$
4. $9\Omega$

Q.

Find the percentage change in the weight of a body when it is taken from equator to poles. The polar radius is 6, 357 km and equatorial radius is 6, 378 km.

Q.

A quantity x is defined as $x=\frac{a^2-b^2}{\sqrt{c+d}}$. Value of a, b, c and d are reported as a = 3 $\pm$0.001, b = 5 $\pm$0.0013, c = 6 $\pm$0.24 and d = 10 $\pm$ 0.4. Percentage error in x will be

1. 2

2. 6

3. 7

4. 4

Q. Percentage errors in the measurement of mass and speed are 2% and 3% respectively. The error in the estimation of kinetic energy obtained by measuring mass and speed will be:

1. 2%
2. 12%
3. 8%
4. 10%

Q. An unknown resistance $X$ is to determined using resistances $R_1,R_2\text{or}{R}_3$. Their corresponding null points are $\text{A},\text{B}$ and $\text{C}$. Which of the above will give the most accurate reading ?
$($Length of $\text{OD}$ is $1\text{m}$. Consider jockey to move gradually on rheostat for changing values)

1. $R=R_1$
2. $R=R_2$
3. $R=R_3$
4. Can't be determined

Q. A physical quantity X is related to four measurable quantities a, b, c and d as given, X = $a^2{b}^3{c}^{5/2}{d}^{-2}$. The percentage error in the measurement of a, b, c and d are 1%, 2%, 2% and 4% respectively. What is the percentage error in quantity X?

1. 15%
2. 17%
3. 21%
4. 23%

Q. A bullet of mass $5\text{g}$, travelling with a speed of $210\text{m/s}$, strikes a fixed wooden target. One half of its kinetic energy is converted into heat in the bullet while the other half is converted into heat in the wood. The rise of temperature of the bullet, if the specific heat of its material is $0.030\text{cal/(g°C)}$ $(1\text{cal}=4.2\times {10}^7\text{ergs}$) close to

1. $87.5°\text{C}$
2. $83.3°\text{C}$
3. $119.2°\text{C}$
4. $38.4°\text{C}$

Q. What is meant by Least Count (LC ) of a measuring instrument?

Q. The least count of a stop watch is 0.2 s. The time of 20 oscillations of a pendulum is measured to be 25 s. The percentage error in the measurement of time will be:

1. 16 %
2. 0.8 %
3. 1.8 %
4. 0.1 %

Q. According to Joule's law of heating, the heat produced is given by $H=I^{2}Rt,$ where $I$ is current, $R$ is resistance and $t$ is the time. If the errors in the measurement of $I,R$ and $t$ are $3\%$, $4\%$ and $6\%$ respectively, then the error in measurement of $H$ is:

1. $8\%$
2. $17\%$
3. $16\%$
4. $25\%$

Q. A sphere of solid material of specific gravity $8$ has a concentric spherical cavity and just sinks in water. Then the ratio of the radius of the cavity to the outer radius of the sphere must be

1. $\frac{\sqrt[3]{33}}{2}$
2. $\frac{\sqrt[3]{35}}{2}$
3. $\frac{\sqrt[3]{37}}{2}$
4. $\frac{3}{\sqrt[3]{37}}$

Q. The relative density of material of a body is found by weighing it first in air and then in water. If the weight in air is $(5.00\pm 0.05)$N and the weight in water is $(4.00\pm 0.05)$N. Find the relative density along with the maximum permissible percentage error.

Q. The relative density of a material of a body is found by weighing it first in air and then in water. If the weight of the body in air is $W_1=8.00\pm 0.05$N and the weight in water is $W_2=6.00\pm 0.05$N, then the relative density ${\rho }_r=\frac{W_1}{W_1-W_2}$ with the maximum permissible error is?

1. $4.00\pm 0.62\%$
2. $4.00\pm 0.82\%$
3. $4.00+3.2\%$
4. $4.00\pm 5.62\%$

Q. 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?

1. $2.5\%$
2. $3.5\%$
3. $4.5\%$
4. $6\%$

Q. Internal electric connections of a multi range voltmeter are shown in the figure. The terminals are marked 3 volt, 15 volt, 150 volt, resistance of the galvanometer is $20\Omega$ and the value of current is 1 mA for the full scale deflection of the galvanometer. The resistance of $R_1$ in $k\Omega$ is-

1. 12
2. 15
3. 3
4. 2.98

Q. In resonance tube experiment we find $l_1=25\text{cm}\text{and}{l}_2=75\text{cm}$. The least count of the scale used to measure $l$ is $0.1\text{cm}$. If there is no error in frequency. What will be the maximum permissiable error in speed of sound. (Take $f_0=325\text{Hz}$ )

1. $2.2\text{m/s}$
2. $1.3\text{m/s}$
3. $2.6\text{m/s}$
4. $0.65\text{m/s}$

Q. A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is $\delta T=0.01$ seconds and he measures the depth of the well to be $L=20$ meters. Take the acceleration due to gravity $g=10m{s}^{-2}$ and the velocity of sound $300m{s}^{-1}$. Then the fractional error in he measurement $\frac{\delta L}{L}$, is closest to

1. $3\%$
2. $0.2\%$
3. $5\%$
4. $0.1\%$

Q. A small filament is at centre of a hollow glass sphere of inner and outer radii 8cm and 9 cm respectively. The refractive index of glass is 1.5 calculate the position of the image of the filament when viewed from outside of the sphere. And also draw the ray diagram