Units & Measurement — JEE Main PYQs

If ε, E and t represent the free space permittivity, electric field and time respectively, then the unit of $\frac{\epsilon E}{t}$ will be :

Keeping the significant figures in view, the sum of the physical quantities $52.01\mathrm{m},153.2\mathrm{m}$ and 0.123 m is :

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 ${\mathrm{v}}_0$. Estimated maximum error in the quantity $\eta$ is : (Ignore errors associated with …

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

In a screw gauge, the zero of the circular scale lies 3 divisions above the horizontal pitch line when their metallic studs are brought in contact. Using this instrument thickness of a sheet is measured. If pitch scale reading is 1 mm and the circular scale reading is 51 then the …

A ball of radius r and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time t, given by $t=A{\rho }^a{r}^b{\eta }^c{\sigma }^d$, where A is a constant and a, b, c and d are integers. The value of $\frac{b+c}{a+d}$ …

When both jaws of vernier callipers touch each other, zero mark of the vernier scale is right to zero mark of main scale, $4^{\text{th}}$ mark on vernier scale coincides with certain mark on the main scale. While measuring the length of a cylinder, observer observes 15 division …

In a vernier callipers, 50 vernier scale divisions are equal to 48 main scale divisions. If one main scale division = 0.05 mm, then the least count of the vernier callipers is ____ mm.

In an experiment, a set of reading are obtained -1.24 mm, 1.25 mm, 1.23 mm, 1.21 mm. The expected least count of the instrument used in recording these readings is ______ mm .

The time period of a simple harmonic oscillator is $\mathrm{T}=2\pi \sqrt{\frac{\mathrm{k}}{\mathrm{m}}}$. The 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 resolut …

In an experiment the values of two spring constants were measured as ${\mathrm{k}}_1=(10\pm 0.2)\mathrm{N}/\mathrm{m}$ and ${\mathrm{k}}_2=(20\pm 0.3)\mathrm{N}/\mathrm{m}$. If these springs are connected in parallel, then the percentage error in equivalent spring con …

Consider a modified Bernoulli equation.

If t has the dimension of time then the dimensions of A and B are ____ , ___\ …

Dimensions of universal gravitational constant (G) in terms of Planck's constant ($h$), distance ($L$), mass ($M$) and time ($T$) are ____.

In a screw gauge the zero of main scale reference line coincides with the fifth division of the circular scale when two studs are in contact. There are 100 divisions in circular scale and pitch of screw gauge is 0.1 mm. When diameter of a sphere is measured, the reading of main s …

In a screw gauge when the circular scale is given five complete rotations it moves linearly by 2.5 mm. If the circular scale has 100 divisions, the least count of screw gauge is ______ mm.

In a Vernier calipers, when both jaws touch each other, zero of the Vernier scale is shifted to the right of zero of the main scale and $7^{\text{th}}$ Vernier division coincides with a main scale reading. If the value of 1 main scale division is 1 mm and there are 10 Vernier sca …

L, C and R represents physical quantities inductance, capacitance and resistance respectively.

The dimensional formula ${\mathrm{ML}}^2{\mathrm{T}}^{-4}{\mathrm{A}}^{-2}$ corresponds to ______.

Match List-I with List-II.

List-I
List-II
(A) Meter (L)
I. $\sqrt{\frac{hc}{\mathrm{G}}}$
(B) Second (S)
II. $\sqrt{\frac{Gh}{{\mathrm{c}}^5}}$
(C) Kilogram (M)
III. $\sqrt{\frac{{\mathrm{K}}^2{\mathrm{L}}^2{\mathrm{c}}^3}{\mathrm{G}h}}$
(D) Kelvin (K)
IV. …

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 measureme …

The density $\rho$ of a uniform cylinder is determined by measuring its mass m, length $l$ and diameter d. The measured values of $\mathrm{m},l$ and d are $97.42\pm 0.02\mathrm{g},8.35\pm 0.05\mathrm{mm}$ and $20.20\pm 0.02\mathrm{mm}$, respectively. Calculat …

The potential energy of a particle changes with distance $x$ from a fixed origin as $V=\frac{A\sqrt{x}}{x+B}$, where A and B are constant with appropriate dimensions. The dimension of AB are ______.

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

A new unit ($\alpha$) of length is chosen such that it is equal to the speed of light in vacuum. What is the distance between Venus and Earth in terms of $\alpha$ units if light takes 6 min. 40 s to cover this distance?

Consider the equation $H=\frac{x^p{ϵ}^q{E}^r}{t^s}$

Where $\mathrm{H}=$ magnetic field; $\mathrm{E}=$ electric field; $ϵ=$ permittivity, $\mathrm{x}=$ distance, $\mathrm{t}=$ time. The values of $p,q,r$ and $s$ respectively are :

The dimensional formula of $\frac{1}{2}{ϵ}_0{\mathrm{E}}^2$ (${ϵ}_0=$ permittivity of vacuum and $\mathrm{E}=$ electric field) is ${\mathrm{M}}^{\mathrm{a}}{\mathrm{L}}^{\mathrm{b}}{\mathrm{T}}^{\mathrm{c}}$. The value of $2\mathrm{a}-\mathrm{b}+\mathrm{c}=$ ____.

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. …

Given below are two statements:

Statement I: In a vernier callipers, one vernier scale division is always smaller than one main scale division.

Statement II: The vernier constant is given by one main scale division multiplied by the number of vernier scale division.

In the lig …

If B is magnetic field and ${\mu }_0$ is permeability of free space, then the dimensions of ($\mathrm{B}/{\mu }_0$) is

The maximum percentage error in the measurment of density of a wire is
[Given, mass of wire $=(0.60\pm 0.003)\mathrm{g}$
radius of wire $=(0.50\pm 0.01)\mathrm{cm}$
length of wire $(10.00\pm 0.05)\mathrm{cm}]$

The position of a particle moving on x -axis is given by $\mathrm{x}(\mathrm{t})=\mathrm{A}\sin \mathrm{t}+\mathrm{B}{\cos }^2\mathrm{t}+{\mathrm{Ct}}^2+\mathrm{D}$, where t is time. The dimension of $\frac{\mathrm{ABC}}{\mathrm{D}}$ is-

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

A physical quantity Q is related to four observables $a,b,c,d$ as follows: $Q=\frac{a{b}^4}{cd}$ where, $\mathrm{a}=(60\pm 3)\mathrm{Pa};\mathrm{b}=(20\pm 0.1)\mathrm{m};\mathrm{c}=(40\pm 0.2){\mathrm{Nsm}}^{-2}$ and $\mathrm{d}=(50\pm 0.1)\mathrm{m}$, then the per …

The pair of physical quantities not having same dimensions is :

The expression given below shows the variation of velocity ( $v$ ) with time ( t ), $v={\mathrm{At}}^2+\frac{\mathrm{Bt}}{\mathrm{C}+\mathrm{t}}$.
The dimension of ABC is :

Match List-I with List-II

List-I
(A) Angular Impulse
(B) Latent Heat
(C) Electrical resistivity
(D) Electromotive force

List-II
(I) $\left[{\mathrm{M}}^0{\mathrm{L}}^2{\mathrm{T}}^{-2}\right]$
(II) $\left[\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-3}{\mathrm{A}}^{-1}\right]$
(II …

A tiny metallic rectangular sheet has length and breadth of 5 mm and 2.5 mm , respectively. Using a specially designed screw gauge which has pitch of 0.75 mm and 15 divisions in the circular scale, you are asked to find the area of the sheet. In this measurement, the maximum frac …

In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of $\left[{\mathrm{M}}^{\mathrm{a}}{\mathrm{L}}^{\mathrm{b}}{\mathrm{T}}^{\mathrm{c}}\right]$. If $b=3$, the value of $c$ is $\_\_\_\_$

For an experimental expression $\mathrm{y}=\frac{32.3\times 1125}{27.4}$, where all the digits are significant. Then to report the value of $y$ we should write :-

The least count of a screw guage is 0.01 mm . If the pitch is increased by $75\%$ and number of divisions on the circular scale is reduced by $50\%$, the new least count will be $\_\_\_\_$ $\times 10^{-3}\mathrm{mm}$.

Given a charge $q$, current $I$ and permeability of vacuum ${\mu }_0$. Which of the following quantity has the dimension of momentum ?

If ${\mu }_0$ and ${\epsilon }_0$ are the permeability and permittivity of free space, respectively, then the dimension of $\left(\frac{1}{{\mu }_0{\epsilon }_0}\right)$ is :

A person measures mass of 3 different particles as $435.42\mathrm{g},226.3\mathrm{g}$ and 0.125 g . According to the rules for arithmetic operations with significant figures, the additions of the masses of 3 particles will be.

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 …

In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of ${\mathrm{M}}^{\mathrm{P}}{\mathrm{L}}^{\mathrm{Q}}{\mathrm{T}}^{\mathrm{R}}{\mathrm{A}}^{\mathrm{S}}$, where value of ' Q ' and ' R ' are

The dimension of $\sqrt{\frac{{\mu }_0}{{ϵ}_0}}$ is equal to that of :
( ${\mu }_0=$ Vacuum permeability and ${ϵ}_0=$ Vacuum permittivity)

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

Match List-I with List-II.

List-I
(A) Coefficient of viscosity
(B) Intensity of wave
(C) Pressure gradient
(D) Compressibility

List-II
(I) $[{\mathrm{ML}}^0{\mathrm{T}}^{-3}]$
(II) $[{\mathrm{ML}}^{-2}{\mathrm{T}}^{-2}]$
(III) $[{\mathrm{M}}^{-1}{\mathrm{LT}}^2]$
(IV) …

Given below are two statements :\\
Statement (I) : Dimensions of specific heat is $\left[{\mathrm{L}}^2{\mathrm{T}}^{-2}{\mathrm{K}}^{-1}\right]$.\\
Statement (II) : Dimensions of gas constant is $\left[{\mathrm{ML}}^2{\mathrm{T}}^{-1}{\mathrm{K}}^{-1}\right]$.\\
In the light of …

In a vernier calliper, when both jaws touch each other, zero of the vernier scale shifts towards left and its $4^{\text{th }}$ division coincides exactly with a certain division on main scale. If 50 vernier scale divisions equal to 49 main scale divisions and zero error in the i …

In finding out refractive index of glass slab the following observations were made through travelling microscope 50 vernier scale division $=49$ MSD; 20 divisions on main scale in each cm For mark on paper\\
$\mathrm{MSR}=8.45\mathrm{cm},\mathrm{VC}=26$\\
For mark on paper see …