Units & Measurement — JEE Main PYQs

In an experiment to find acceleration due to gravity $(g)$ using simple pendulum, time period of 0.5 s is measured from time of 100 oscillations with a watch of 1 s resolution. If measured value of length is 10 cm known to 1 mm accuracy, the accuracy in the determination of $g$ i …

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 one division of main scale of vernier callipers reads 1 mm and 10 divisions of Vernier scale is equal to the 9 divisions on main scale. When the two jaws of the instrument touch each other the zero of the Vernier lies to the right of zero of the main scale and its fourth divi …

If momentum $[\mathrm{P}]$, area $[\mathrm{A}]$ and time $[\mathrm{T}]$ are taken as fundamental quantities, then the dimensional formula for coefficient of viscosity is :

Which of the following physical quantities have the same dimensions?

An expression of energy density is given by $\mathrm{u}=\frac{\alpha }{\beta }\sin \left(\frac{\alpha \mathrm{x}}{\mathrm{kt}}\right)$, where $\alpha ,\beta$ are constants, x is displacement, k is Boltzmann constant and t is the temperature. The dimensions of $\beta$ will be …

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:

A screw gauge of pitch 0.5 mm is used to measure the diameter of uniform wire of length 6.8 cm , the main scale reading is 1.5 mm and circular scale reading is 7 . The calculated curved surface area of wire to appropriate significant figures is :
[Screw gauge has 50 divisions on …

In an experiment of determine the Young's modulus of wire of a length exactly 1 m , the extension in the length of the wire is measured as 0.4 mm with an uncertainty of $\pm 0.02\mathrm{mm}$ when a load of 1 kg is applied. The diameter of the wire is measured as 0.4 mm with an …

Velocity (v) and acceleration (a) in two systems of units 1 and 2 are related as ${\mathrm{v}}_2=\frac{\mathrm{n}}{{\mathrm{m}}^2}{\mathrm{v}}_1$ and ${\mathrm{a}}_2=\frac{{\mathrm{a}}_1}{\mathrm{mn}}$ respectively. Here m and n are constants. The relations for distance and …

A student in the laboratory measures thickness of a wire using screw gauge. The readings are 1.22 $\mathrm{mm}$, 1.23 $\mathrm{mm}$, 1.19 $\mathrm{mm}$ and 1.20 mm. The percentage error is $\frac{\mathrm{x}}{121}\%$. The value of x is

The SI unit of a physical quantity is pascal-second. The dimensional formula of this quantity will be

The distance of the Sun from earth is $1.5\times 10^{11}\mathrm{m}$ and its angular diameter is (2000) s when observed from the earth. The diameter of the Sun will be :

If $\mathrm{L}$, $\mathrm{C}$ and R are the self inductance, capacitance and resistance respectively, which of the following does not have the dimension of time?

The vernier constant of Vernier callipers is 0.1 mm and it has zero error of $(-0.05)\mathrm{cm}$. While measuring diameter of a sphere, the main scale reading is 1.7 cm and coinciding vernier division is 5. The corrected diameter will be ____ $\times 10^{-2}\mathrm{cm}$.

A travelling microscope is used to determine the refractive index of a glass slab. If 40 divisions are there in 1 cm on main scale and 50 Vernier scale divisions are equal to 49 main scale divisions, then least count of the travelling microscope is ______ $\times 10^{-6}$ m.

In Vander Waals equation $\left[P+\frac{a}{V^2}\right][V-b]=RT$; P is pressure, V is volume, R is universal gas constant and T is temperature. The ratio of constants $\frac{a}{b}$ is dimensionally equal to:

An expression for a dimensionless quantity P is given by $\mathrm{P}=\frac{\alpha }{\beta }{\log }_{\mathrm{e}}\left(\frac{\mathrm{kt}}{\beta \mathrm{x}}\right)$; where $\alpha$ and $\beta$ are constants, x is distance; k is Boltzmann constant and $t$ is the temperature. The …

In a vernier callipers, each cm on the main scale is divided into 20 equal parts. If tenth vernier scale division coincides with nineth main scale division. Then the value of vernier constant will be ____ $\times 10^{-2}\mathrm{mm}$.

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

Given below are two statements : One is labelled as Assertion (A) and other is labelled as Reason (R).
Assertion (A) : Time period of oscillation of a liquid drop depends on surface tension (S), if density of the liquid is $p$ and radius of the drop is $r$, then …

A travelling microscope has 20 divisions per cm on the main scale while its Vernier scale has total 50 divisions and 25 Vernier scale divisions are equal to 24 main scale divisions, what is the least count of the travelling microscope ?

In an experiment to find out the diameter of wire using screw gauge, the following observation were noted :

(a) Screw moves 0.5 mm on main scale in one complete rotation
(b) Total divisions on circular scale $=50$
(c) Main scale reading is 2.5 mm
(d) $45^{\text{th }}$ divis …

Identify the pair of physical quantities which have different dimensions :

Identify the pair of physical quantities that have same dimensions :

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 ____%.

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

In a Screw Gauge, fifth division of the circular scale coincides with the reference line when the ratchet is closed. There are 50 divisions on the circular scale, and the main scale moves by 0.5 mm on a complete rotation. For a particular observation the reading on the main scale …

If $\mathrm{E},\mathrm{L},\mathrm{M}$ and G denote the quantities as energy, angular momentum, mass and constant of gravitation respectively, then the dimensions of $P$ in the formula $P=E{L}^2{M}^{-5}{G}^{-2}$ are :-

The work done by a gas molecule in an isolated system is given by $W=\alpha {\beta }^2{e}^{-\frac{x^2}{\alpha kT}}$, where $x$ is the displacement, $k$ is the Boltzmann constant and $T$ is the temperature. $\alpha$ and $\beta$ are constants. Then the dimensions of $\beta$ will b …

Match List-I with List-II:

List - I
(a) Magnetic Induction
(b) Magnetic Flux
(c) Magnetic Permeability
(d) Magnetization

List - II
(i) ${\mathrm{ML}}^2{\mathrm{T}}^{-2}{\mathrm{A}}^{-1}$
(ii) ${\mathrm{M}}^0{\mathrm{L}}^{-1}\mathrm{A}$
(iii) $M{T}^{-2}{A}^{-1}$
(iv) …

The acceleration due to gravity is found upto an accuracy of $4\%$ on a planet. The energy supplied to a simple pendulum of 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 …

The entropy of any system is given by $\mathrm{S}={\alpha }^2\beta \ell \mathrm{n}\left[\frac{\mu \mathrm{kR}}{\mathrm{J}{\beta }^2}+3\right]$ Where $\alpha$ and $\beta$ are the constants. $\mu ,\mathrm{J},\mathrm{k}$ and R no. of moles, mechanical equivalent of heat, Boltzmann …

The period of oscillation of a simple pendulum is $T=2\pi \sqrt{\frac{L}{g}}$. Measured value of ' $L$ ' is 1.0 m from meter 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 …

If time ( t ), velocity ( u ), and angular momentum ( $\ell$ ) are taken as the fundamental units. Then the dimension of mass $(\mathrm{m})$ in terms of $\mathrm{t},\mathrm{v}$, and $\ell$ is :

If force (F), length (L) and time (T) are taken as the fundamental quantities. Then what will be the dimension of density :

The pitch of the screw gauge is 1 mm and there are 100 divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies 8 divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is cl …

Match List - I with List - II :

List - I
(a) h (Planck's constant)
(b) E (kinetic energy)
(c) V (electric potential)
(d) P (liner momentum )

List - II
(i) $\left[\mathrm{ML}{\mathrm{T}}^{-1}\right]$
(ii) $\left[\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-1}\right]$
(ii …

Three students $S_1,S_2$ and $S_3$ 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.

| …

In a typical combustion engine the work done by a gas molecule is given by $\mathrm{W}={\alpha }^2\beta {\mathrm{e}}^{-\frac{\beta {\mathrm{x}}^2}{\mathrm{kT}}}$, where $x$ is the displacement, $k$ is the Boltzmann constant and $T$ is the temperature. If $\alpha$ and …

Which of the following is not a dimensionless quantity?

If '$C$' and '$V$' represent capacity and voltage respectively then what are the dimensions of $\lambda$ where $\mathrm{C}/\mathrm{V}=\lambda$?

If velocity $[\mathrm{V}]$, time $[\mathrm{T}]$ and force $[\mathrm{F}]$ are chosen as the base quantities, the dimensions of the mass will be :

The diameter of a spherical bob is measured using a vernier callipers. 9 divisions of the main scale, in the vernier callipers, are equal to 10 divisions of vernier scale. One main scale division is 1 mm. The main scale reading is 10 mm and $8^{\text{th}}$ division of vernier sca …

Which of the following equations is dimensionally incorrect ?
Where $\mathrm{t}=$ time, $\mathrm{h}=$ height, $\mathrm{s}=$ surface tension, $\theta =$ angle, $\mathrm{p}=$ density, $\mathrm{a},\mathrm{r}=$ radius, $\mathrm{g}=$ acceleration due to gravity, …

Match List-I with List-II.

List-I:
(a) Torque
(b) Impulse
(c) Tension
(d) Surface Tension

List-II:
(i) $ML{T}^{-1}$
(ii) $M{T}^{-2}$
(iii) $M{L}^2{T}^{-2}$
(iv) $ML{T}^{-2}$

Choose the most appropriate answer from the option given below :

Assertion A: If in five complete rotations of the circular scale, the distance travelled on main scale of the screw gauge is 5 mm and there are 50 total divisions on circular scale, then least count is 0.001 cm.

Reason R: Least Count …

The vernier-scale used for measurement has a positive zero error of 0.2 mm. If while taking a measurement it was noted that '0' on the vernier-scale lies between 8.5 cm and 8.6 cm, vernier coincidence is 6, then the correct value of measurement is (least count $=0.01$ cm):

Student A and Student B used two screw gauges of equal pitch and 100 equal circular divisions to measure the radius of a given wire. The actual value of the radius of the wire is 0.322 cm. The absolute value of the difference between the final circular scale readings observed by …

The force is given in terms of time $t$ and displacement $x$ by the equation
$\mathrm{F}=\mathrm{A}\cos \mathrm{Bx}+\mathrm{C}\sin \mathrm{Dt}$
The dimensional formula of $\frac{AD}{B}$ is :