Units & measurements practice · Mixed difficulty

Which of the following statement(s) is/are incorrect?

Identify the pair whose dimensions are equal.

In a Vernier Calipers, 10 divisions of Vernier scale is equal to the 9 divisions of main scale. When both jaws of Vernier calipers touch each other, the zero of the Vernier scale is shifted to the left of zero of the main scale and $4^{\text {th }}$ Vernier scale division exactly coincides with the main scale reading. One main scale division is equal to $1 \mathrm{~mm}$. While measuring diameter of a spherical body, the body is held between two jaws. It is now observed that zero of the Vernier scale lies between 30 and 31 divisions of main scale reading and $6^{\text {th }}$ Vernier scale division exactly coincides with the main scale reading. The diameter of the spherical body will be

$S = A(1 - {e^{ - Bxt}})$ where S is speed and x is displacement. Then unit of B is

$\left[ {M{L^2}{T^{ - 3}}} \right]$ is the dimension of

Assertion: ‘Light year’ and ‘Wavelength’ both measure
distance.
Reason : Both have dimension of time.

A screw gauge has 50 divisions on its circular scale. The circular scale is 4 units ahead of the pitch scale marking, prior to use. Upon one complete rotation of the circular scale, a displacement of 0.5 mm is noticed on the pitch scale. The nature of zero error involved and the least count of the screw gauge, are respectively.

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

Which physical quantities have same dimensions?

The dimensions of length are expressed as $G^x C^y h^z$. $\mathrm{G}$ is the universal gravitational constant, $\mathrm{c}$ is the speed of light and $\mathrm{h}$ is the Planck's constant. The value of $\left(\frac{x-y}{z}\right)^{\frac{1}{x}}$ is

The unit "henry" is equivalent to:

Dimensions of charge are

The screw gauge has least count equal to $0.02 \mathrm{~mm}$ and there are 50 divisions in its circular scales. The pitch of the screw gauge is

Assertion: Avogadro number is the number of atoms in one gram mole.
Reason: Avogadro number is a dimensionless constant.

The dimensions of $\left(\mu_0 \varepsilon_0\right)^{-\frac{1}{2}}$ are

The current flowing through a resistor $10.932\, \mathrm{ohm}$ is $4.25\, \mathrm{amp}$. The potential difference is 46.461 volt. The potential in significant figures is

Temperature can be expressed as a derived quantity in terms of any of the following

The velocity $\mathrm{v}$ of a particle at time $\mathrm{t}$ is given by $\mathrm{v}=\mathrm{at}+\frac{\mathrm{b}}{\mathrm{t}+\mathrm{c}}$, where $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are constants. The dimensions of $\mathrm{a}$, $\mathrm{b}$ and $\mathrm{c}$ are respectively:

How many wavelengths of $K{r^{86}}\mathrm{\;}$ are there in one metre

A unit of area, often used in measuring land areas, is the hectare defined as ${10^4}\,{m^2}$. An open-pit coal mine consumes 75 hectares of land, down to a depth of 26m, each year. What volume of earth, in cubic kilometre, is removed in this time?

Statement-I: Systematic errors can be removed completely.
Statement-II: the cause of systematic errors can be known.

The dimensional formula of coefficient of permittivity for free space $\left( {{\varepsilon _0}} \right)$ in the equation $F = \frac{1}{{4\mathrm{\pi }{\varepsilon _0}}}\frac{{{q_1}{q_2}}}{{{r^2}}},$ where symbols have their usual meanings, is

The physical quantity that has the same dimensional formula as pressure is:

The relative error in calculating the value of $g$ from the relation $T=2 \pi \sqrt{\frac{l}{g}}$ is (given the relative errors in calculating $\mathrm{T}$ and $l$ are $\pm x$ and $\pm y$ respectively)

The dimensional formula for impulse is

Surface tension has the same dimensions as that of

Dimensional formula of magnetic flux is

Statement-I: The length of an object is measured with two instruments as $l=4.01 \mathrm{~cm}$ and $l=4.009 \mathrm{~cm}$. The second instrument has a better resolution.
Statement-II: More value is the least count of an instrument, better is the resolution.

Which physical quantities have the same dimension

The dimensional formula for Reynold's number is