Units & Measurement — JEE Advanced PYQs

Two thin wires, Wire-1 of diameter 0.650 mm and Wire-2 of unknown diameter $d$ are given. To obtain the value of $d$, the diameters of the two wires are measured with a screw gauge. The screw gauge has a pitch of 0.5 mm and there are 100 divisions on the circular scale (CS). …

In a new system of units, the units of mass, length, time and current are 5 kg, 5 m, 5 s and 5 A, respectively. If ${\mu }_0$ and ${ϵ}_0$ are the permeability and permittivity of free space, respectively, then in this new system of units, the magnitude of one SI unit of …

Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter $D$ of a tube. The measured value of $D$ is:

Length, breadth and thickness of a strip having a uniform cross section are measured to be 10.5 cm, 0.05 mm, and $6.0\mu \mathrm{m}$, respectively. Which of the following option(s) give(s) the volume of the strip in ${\mathrm{cm}}^3$ with correct significant figures:

A single slit diffraction experiment is performed to determine the slit width using the equation, $\frac{bd}{D}=m\lambda$, where $b$ is the slit width, $D$ the shortest distance between the slit and the screen, $d$ the distance between the $m^{\text{th }}$ diffractio …

A temperature difference can generate e.m.f. in some materials. Let $S$ be the e.m.f. produced per unit temperature difference between the ends of a wire, $\sigma$ the electrical conductivity and $\kappa$ the thermal conductivity of the material of the wire. Taking …

A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space ${\epsilon }_0$, Planck's constant h, and speed of light $c$. If the dimensionless quantity is written as …

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 ____.

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

Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$, Planck's constant $h$ and the speed of light c , as $Y={\mathrm{c}}^{\alpha }{h}^{\beta }{G}^{\gamma }$. Which of the following is the correct option ?

In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$, electron mass $m_e$, Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4\pi {ϵ}_0}$, where ${ϵ}_0$ is the permittivity of vacuum. In terms …

Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is 0.5 mm. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.

| Mea …

A physical quantity $\vec{S}$ is defined as $\vec{S}=(\vec{E}\times \vec{B})/{\mu }_0$, where $\vec{E}$ is electric field, $\vec{B}$ is magnetic field and ${\mu }_0$ is the permeability of free space. The dimensions of $\vec{S}$ are the same as the dimensions of whic …

The smallest division on the main scale of a Vernier calipers is 0.1 cm. Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this calipers with no gap between its two jaws. The figure on the right sh …

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity X as follows: [position] $=\left[X^{\alpha }\right]$; [ …

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 i …

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of L, which of the following statement (s) is/are correct ?

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $[E]$ and $[B]$ stand for dimensions of electric and magnetic fi …

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $[E]$ and $[B]$ stand for dimensions of electric and magnetic fi …

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 relatio …

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 $L=20$ mete …

A length-scale $(l)$ depends on the permittivity $(\epsilon )$ of a dielectric material, Boltzmann constant $(k_B)$, the absolute temperature $(T)$, the number per unit volume $(n)$ of certain charged particles, and the charge $(q)$ carried by each of the particle …

There are two Vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale. The Vernier scale of one of the calipers ( $C_1$ ) has 10 equal divisions that correspond to 9 main scale divisions. The Vernier scale of the other caliper …

In an experiment to determine the acceleration due to gravity $g$, the formula used for the time period of a periodic motion is $T=2\pi \sqrt{\frac{7(R-r)}{5g}}$. The values of $R$ and $r$ are measured to be $(60\pm 1)\mathrm{mm}$ and $(10\pm 1)\mathrm{mm}$, res …

Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier scale coincide with 4 divisions on the main scale and in the screw …

Planck's constant h , speed of light c and gravitational constant G are used to from a unit of length L and a unit of mass M . Then the correct options(s) is (are)

The energy of a system as a function of time $t$ is given as $E(t)=A^2\exp (-\alpha t)$, where $\alpha =0.2{\mathrm{s}}^{-1}$. 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 …

In terms of potential difference V, electric current I, permittivity ${\epsilon }_0$, permeability ${\mu }_0$ and speed of light c, the dimensionally correct equation(s) is(are)

To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density $\rho$ of the fog, intensity (power/area) $S$ of the light from the signal and its f …

The diameter of a cylinder is measured using a Vernier callipers with no zero error. It is found that the zero of the Vernier scale lies between $5.10cm$ and $5.15cm$ of the main scale. The Vernier scale has 50 divisions equivalent to $2.45cm$. The …

Using the expression $2d\sin \theta =\lambda$, one calculates the values of $d$ by measuring the corresponding angles $\theta$ in the range 0 to 90°. The wavelength $\lambda$ is exactly known and the error in $\theta$ is constant for all values of $\theta$. As …

Match List I with List II and select the correct answer using the codes given below the lists :
| 1. | …

In the determination of Young's modulus $\left(Y=\frac{4ML\mathrm{g}}{\pi l{d}^2}\right)$ by using Searle's method, a wire of length $L=2\mathrm{m}$ and diameter $d=0.5\mathrm{mm}$ is used. For a load $M=2.5\mathrm{kg}$, an extension $l=0.25\mathrm{mm}$ in …

A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let 'N' be the number density of free electrons, each of mass 'm'. When the ele …

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 division …

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

A Vernier calipers has 1 mm marks on the main scale. It has 20 equal divisions on the Vernier scale which match with 16 main scale divisions. For this Vernier calipers, the least count is

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 f …

Some physical quantities are given in Column I and some possible SI units in which these quantities may be expressed are given in Column II. Match the physical quantities in Column I with the units in Column II and indicate your answer by darkening appropriate bubbles in the 4 × …

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 ±0.05 mm at a load of exactly 1.0 kg. The s …

A student performs an experiment for determination of $\mathrm{g}\left(=\frac{4{\pi }^2\ell }{{\mathrm{T}}^2}\right),\ell \approx 1\mathrm{m}$, and he commits an error of $\Delta \ell$. For T he takes the time of n oscillations with the stop watch of least count …

The circular divisions of shown screw gauge are 50. It moves 0.5 mm on main scale in one rotation. The diameter of the ball is