Errors in Measurement

Four persons measure the length of a rod as $20.00\text{ cm}, 19.75\text{ cm}, 17.01\text{ cm}$ and 18.25 cm. The relative error in the measurement of average length of the rod is:

If two resistors of resistances have values $R_1 = (350 \pm 3)\ \Omega$ and $R_2 = (140 \pm 4)\ \Omega$. The percentage error for the sum and difference of $R_1$ and $R_2$ are respectively

Mass and volume of a body are found to be $(6.00 \pm 0.03)$ kg and $(2.00 \pm 0.02)$ m$^{3}$ respectively. Then the density of the body is

A wire has mass $(0.4 \pm 0.002)$ g, radius $(0.6 \pm 0.006)$ mm and length $(8 \pm 0.04)$ cm. The percentage error in the measurement of its density is

In a certain system, the quantities A and B are measured as $8 \pm 0.08$ and $6 \pm 0.06$ respectively. Let $x = AB,\ y = A + B$ and $z = A - B$. The correct order of % error in $x,\ y$ and $z$ is

If the error in the measurement of volume of the sphere is $6\%$ then the error in the measurement of its surface area will be

The resistance of a conductor is $R = V/I$ where voltage $V = 100 \pm 5$ V and current $I = 10 \pm 0.2$ A. The error in the calculation of $R$ is

The percentage error in the measurement of mass and speed are $2\%$ and $3\%$ respectively. The maximum error in the estimation of kinetic energy is

A quantity $Q$ is formulated $X^{-2}Y^{+\frac{3}{2}}Z^{-\frac{2}{5}}.\ X,\ Y$ 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

A physical quantity $x$ is related as $x = \sqrt{a}\,b^{2}c^{3}d^{-4}$. Relative errors in the quantities $a,\ b,\ c$ and $d$ are 2%, 1%, 3% and 4% respectively. Relative error in $x$ will be

The voltage across a lamp is $(6.0 \pm 0.3)$ volt and the current passing through it is $(4.0 \pm 0.1)$ ampere. The power consumed in watt will be (using percentage error)

Mass and volume of a body are found to be $(6.00 \pm 0.03)$ kg and $(2.00 \pm 0.02)$ m$^{3}$ respectively. Then the density of a body is

The error in the measurement of length and mass of a cube is 3% and 4% respectively. The error in the measurement of its density will be

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

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

A quantity P is related as $X^{-2}Y^{-3/2}Z^{2/5}$ where X,Y,Z are independent parameters which have fractional errors of $0.1, 0.2$ and $0.5$ respectively in measurement. The maximum fractional error in P is

The percentage error in the calculated volume of a sphere, if there is $2\%$ error in its diameter measurement, is ______.

In an experiment the values of two spring constants were measured as $k_1 = (10 \pm 0.2)\text{ N/m}$ and $k_2 = (20 \pm 0.3)\text{ N/m}$. If these springs are connected in parallel, then the percentage error in equivalent spring constant is:

A physical quantity $C$ is related to four other quantities $P,\ Q,\ R$ and $S$ as $C = \frac{PQ^2}{R^3\sqrt{S}}$. The percentage error 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

The period of oscillation of a simple pendulum is $T = 2\pi\sqrt{\frac{l}{g}}$ where $l$ is about $100$ cm and has $1$ mm accuracy. The period is about $2$ s. The time of $100$ oscillations is measured by a stopwatch of least count $0.1$ s. The percentage error in '$g$' is (gravitation acceleration)

A particular physical quantity is given by $X = 2\pi\sqrt{\frac{a}{b}}$ and the measured value of 'a' is $40$ unit with $0.01$ unit accuracy. The value of 'b' is $\frac{98}{10}$ unit with $1$ unit accuracy. The % error in $X$ is nearly

The percentage error in the measurement of mass of a body and its speed are $0.73\%$ and $1.84\%$ respectively. The percentage error in the measurement of its momentum and kinetic energy are respectively

A cuboid has volume $V = l \times l \times \sqrt{l}$, where $l$ is the length of one side. When the length $l$ is measured by a meter scale of least count $1$ mm, the relative percentage error in the measurement of its volume is $6.25\%$. Then the relative percentage error in measurement of its length and the value of the length $l$ is

The maximum error in the measurement of the density and mass of the uniform cube are 8% and 2% respectively. Hence, the maximum error in the measurement of length will be

A force 'F' is applied on a square plate of side 'L'. If the percentage error in determining 'L' is 3% and that in 'F' is 2% then the percentage error in determining the pressure is

The length of rod is measured by meter scale having least count $0.1$ cm and diameter is measured by vernier callipers having least count $0.01$ cm. Length of rod is $5.0$ cm and radius $2.0$ cm. The percentage error in the calculated value of the volume is

The length of a cylinder is measured with a meter rod having least count $0.1$ cm. Its diameter is measured with vernier calipers having least count $0.01$ cm. Length of cylinder is $8.0$ cm and radius is $4.0$ cm. Find the percentage error in the calculated value of the volume.

The period of oscillation of a simple pendulum is $T = 2\pi(L/g)^{1/2}$. Measured value of 'L' is 10 cm known to 1 mm accuracy and time for 100 oscillations is 50 second, using a watch of 1 second resolution. The percentage error in the measurement of 'g' is

The period of oscillation of a simple pendulum is given by $T = 2\pi\sqrt{\frac{l}{g}}$ where $l$ is about $80$ cm and is known to have $0.1$ cm accuracy. The period is about $1.5$ s. The time of $50$ oscillations is measured by a stop watch of least count $0.1$ s. The percentage error in $g$ is nearly

The density $\rho$ of a uniform cylinder is determined by measuring its mass $m$, length $l$ and diameter $d$. The measured values of $m, l$ and $d$ are $97.42 \pm 0.02\text{ g},\ 8.35 \pm 0.05\text{ mm}$ and $20.20 \pm 0.02\text{ mm}$, respectively. Calculated percentage fractional error in $\rho$ is ______.

The speed of a longitudinal wave in a metallic bar is 400 m/s. If the density and Young's modulus of the bar material are increased by 0.5% and 1%, respectively then the speed of the wave is changed approximately to ______ m/s.

A spherical body of radius $r$ and density $\sigma$ falls freely through a viscous liquid having density $\rho$ and viscosity $\eta$ and attains a terminal velocity $v_0$. Estimated maximum error in the quantity $\eta$ is: (Ignore errors associated with $\sigma,\ \rho$ and $g$, gravitational acceleration)

In an experiment to determine the resistance of a given wire using Ohm's law, the voltmeter and ammeter readings are noted as 10 V and 5 A, respectively. The least counts of voltmeter and ammeter are 500 mV and 200 mA, respectively. The estimated error in the resistance measurement is $\underline{\qquad}\ \Omega$

The diameter of a wire measured by a screw gauge of least count 0.001 cm is 0.08 cm. The length measured by a scale of least count 0.1 cm is 150 cm. When a weight of 100 N is applied to the wire, the extension in length is 0.5 cm, measured by a micrometer of least count 0.001 cm. The error in the measured Young’s modulus is $\alpha \times 10^{9}\text{ N/m}^{2}$. The value of $\alpha$ is ________.

(Ignore the contribution of the load to Young’s modulus error calculation)

The time period of a simple harmonic oscillator is $T = 2\pi\sqrt{\frac{m}{k}}$. Measured value of mass $(m)$ of the object is 10 g with an accuracy of 10 mg and time for 50 oscillations of the spring is found to be 60 s using a watch of 2 s resolution. Percentage error in determination of spring constant $(k)$ is ________%.

Which of the following comes under the category of random errors?

The initial and final temperatures of water as recorded by an observer are $(38.6 \pm 0.2)^\circ\text{C}$ and $(82.3 \pm 0.3)^\circ\text{C}$. The rise in temperature with proper error limits is

In an experiment four quantities $p, q, r$ and s are measured with percentage $3\%, 2\%, 3\%$ and $1\%$ respectively. Quantity ' $A$ ' is calculated as follows

$A = \frac{pq^{2}}{r^{2}s^{4}}$, the percentage error in ' A ' is

The error in the measurement of length and mass of a cube is $3\%$ and $4\%$ respectively. The error in the measurement of density will be

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

The percentage error in the measurement of mass and speed of a particular body is $3\%$ and $4\%$ respectively. The percentage error in the measurement of kinetic energy is

Error in the measurement of radius of the sphere is $2\%$. The error in the calculated value of its volume is

The maximum percentage error in the measurement of density of a wire is

[Given, mass of wire $= (0.60 \pm 0.003)\text{ g}$

radius of wire $= (0.50 \pm 0.01)\text{ cm}$

length of wire $= (10.00 \pm 0.05)\text{ cm}]$

A student measures time for 20 oscillations of a simple pendulum as $30\text{ s}, 32\text{ s}, 35\text{ s}$ and 35 s. If the minimum division in the measuring clock is 1 s, then correct mean time (in second) is

A physical quantity $P$ is related to four observations $a, b, c$ and $d$ as follows:

$P = a^3b^2/c\sqrt{d}$

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

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

A force F is applied on a square plate of side L. If the percentage error in determining F is $3\%$ and that in L is $2\%$, then the percentage error in determining the pressure is

The pressure on a square plate is measured by measuring the force acting on the plate and length of the sides of the plate. The maximum error in the measurement of force and length are respectively $4\%$ and $2\%$, the percentage error in the measurement of pressure is

The period of oscillating simple pendulum is $T = 2\pi\sqrt{\frac{l}{g}}$ where length ' $l$ ' is 100 cm with error 1 mm. Period is 2 second. The time of 100 oscillations is measured by a stopwatch of least count 0.1s. The percentage error in gravitational acceleration ' g ' is

A physical quantity A can be determined by measuring parameters $B, C, D$ and E using the relation $A = \frac{B^{\alpha} C^{\beta}}{D^{\gamma} E^{\delta}}$. If the maximum errors in the measurement are $b\%,\ c\%,\ d\%$ and $e\%$ then maximum error in the value of A is

The density of a cube is measured by measuring its mass and length of its sides. The \% error in the measurement of mass and length are $5\%$ and $6\%$ respectively. The percentage error in the measurement of density is

The formula for the physical quantity is $P = \frac{x^{3} y}{z^{2}}$ and the percentage error in the determination of physical quantities $x, y, z$ are $0.6\%,\ 3\%$ and $1.3\%$ respectively. The percentage error in the measurement of P is

A quantity Q is formulated as $X^{-2} Y^{+\frac{3}{2}} Z^{-\frac{2}{5}}$. 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

The energy of a system is given as $E(t) = \alpha^{3} e^{-\beta t}$, where t is the time and $\beta = 0.3\text{ s}^{-1}$. The errors in the measurement of $\alpha$ and $t$ are $1.2\%$ and $1.6\%$, respectively. At $t = 5\text{ s}$, maximum percentage error in the energy is:

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

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

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__________%

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

$Q = \frac{ab^{4}}{cd}$

where, $a = (60 \pm 3)\text{ Pa};\ b = (20 \pm 0.1)\text{ m};\ c = (40 \pm 0.2)\text{ Nsm}^{-2}$ and $d = (50 \pm 0.1)\text{ m}$, then the percentage error in Q is $\frac{x}{1000}$, where $x =$ _________.

In an electrical circuit, the voltage is measured as $V = (200 \pm 4)$ volt and the current is measured as $I = (20 \pm 0.2)$ A. The value of the resistance is:

The radius $(r)$, length $(l)$ and resistance $(R)$ of a metal wire was measured in the laboratory as $r = (0.35 \pm 0.05)\text{ cm}$

$R = (100 \pm 10)\text{ ohm}$

$l = (15 \pm 0.2)\text{ cm}$

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

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:

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

The resistance $R = \frac{V}{I}$ where $V = (200 \pm 5)\text{ V}$ and $I = (20 \pm 0.2)\text{ A}$, the percentage error in the measurement of $R$ is:

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:

In an experiment to measure focal length ($f$) of convex lens, the least counts of the measuring scales for the position of object (u) and for the position of image (v) are $\Delta u$ and $\Delta v$, respectively. The error in the measurement of the focal length of the convex lens will be:

The measured value of the length of a simple pendulum is $20\text{ cm}$ with $2\text{ mm}$ 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:

The errors in the measurement which arise due to unpredictable fluctuations in temperature and voltage supply are:

The diameter of a spherical bob, when measured with vernier callipers yielded the following values: $3.33\text{ cm}, 3.32\text{ cm}, 3.34\text{ cm}, 3.33\text{ cm}$ and $3.32\text{ cm}$. The mean diameter to appropriate significant figures is:

Two resistances are given as $R_1 = (10 \pm 0.5)\ \Omega$ and $R_2 = (15 \pm 0.5)\ \Omega$. The percentage error in the measurement of equivalent resistance when they are connected in parallel is -

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

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

A cylindrical wire of mass $(0.4 \pm 0.01)\text{ g}$ has length $(8 \pm 0.04)\text{ cm}$ and radius $(6 \pm 0.03)\text{ mm}$. The maximum error in its density will be:

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

The percentage error in the measurement of g is: (Given that $g = \frac{4\pi^2 L}{T^2},\ L = (10 \pm 0.1)$ cm, $T = (100 \pm 1)$ s)

A silver wire has a mass (0.6 $\pm$ 0.006) g, radius (0.5 $\pm$ 0.005) mm and length (4 $\pm$ 0.04) cm. The maximum percentage error in the measurement of its density will be:

If $Z = \frac{A^2B^3}{C^4}$, then the relative error in Z will be:

A torque meter is calibrated to reference standards of mass, length and time each with $5\%$ accuracy. After calibration, the measured torque with this torque meter will have net accuracy of:

The maximum error in the measurement of resistance, current and time for which current flows in an electrical circuit are $1\%,\ 2\%$ and $3\%$ respectively. The maximum percentage error in the detection of the dissipated heat will be:

For $z = a^{2} x^{3} y^{\frac{1}{2}}$, where 'a' is a constant. If percentage error in measurement of 'x' and 'y' are 4% and 12% respectively, then the percentage error for 'z' will be _______________%.

In an experiment to find acceleration due to gravity (g) using simple pendulum, time period of $0.5\text{ s}$ is measured from time of 100 oscillation with a watch of $1\text{ s}$ resolution. If measured value of length is $10\text{ cm}$ known to $1\text{ mm}$ accuracy, The accuracy in the determination of $g$ is found to be $x\%$. The value of $x$ is ___________.

If the length of the pendulum in pendulum clock increases by 0.1%, then the error in time per day is:

A physical quantity 'y' is represented by the formula $y = m^{2} r^{-4} g^{x} l^{-\frac{3}{2}}$

If the percentage errors found in y, m, r, l and g are 18, 1, 0.5, 4 and p respectively, then find the value of x and p.

The radius of a sphere is measured to be (7.50 $\pm$ 0.85) cm. Suppose the percentage error in its volume is x.

The value of x, to the nearest integer, is __________.

The resistance R = $\frac{V}{I}$, where V = (50 $\pm$ 2)V and I = (20 $\pm$ 0.2)A. The percentage error in R is 'x'%. The value of 'x' to the nearest integer is _________.

A student determined Young's Modulus of elasticity using the formula $Y = \frac{MgL^{3}}{4bd^{3}\delta}$. The value of g is taken to be 9.8 m/s2, without any significant error, his observations are as follows.

Then the fractional error in the measurement of Y is:

Three students S1, S2 and S3 perform an experiment for determining the acceleration due to gravity (g) using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.

(Least count of length = 0.1 cm and Least count for time = 0.1 s)

If E1, E2 and E3 are the percentage errors in 'g' for students 1, 2 and 3 respectively, then the minimum percentage error is obtained by student no. ______________.

In the experiment of Ohm's law, a potential difference of 5.0 V is applied across the end of a conductor of length 10.0 cm and diameter of 5.00 mm. The measured current in the conductor is 2.00 A. The maximum permissible percentage error in the resistivity of the conductor is:

The time period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{l}{g}}$. The measured value of the length of pendulum is 10 cm known to a 1mm accuracy. The time for 200 oscillations of the pendulum is found to be 100 second using a clock of 1s resolution. The percentage accuracy in the determination of 'g' using this pendulum is 'x'. The value of 'x' to be nearest integer is :-

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 1m (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?