Units, Dimensions and Errors

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 $\mu$m, respectively. Which of the following option(s) give(s) the volume of the strip in cm$^3$ with correct significant figures:

A single slit diffraction experiment is performed to determine the slit width using the equation, $\dfrac{bd}{D} = m\lambda$, where $b$ is the slit width, $D$ the shortest distance between the slit and the screen, $d$ the distance between the $m^{\text{th}}$ diffraction maximum and the central maximum, and $\lambda$ is the wavelength. $D$ and $d$ are measured with scales of least count of 1 cm and 1 mm, respectively. The values of $\lambda$ and $m$ are known precisely to be 600 nm and 3, respectively. The absolute error (in $\mu$m) in the value of $b$ estimated using the diffraction maximum that occurs for $m = 3$ with $d = 5$ mm and $D = 1$ m is _____.

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 $M, L, T, I$ and $K$ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $Z = \dfrac{S^2\sigma}{\kappa}$ is:

A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space $\varepsilon_0$, Planck's constant $h$, and speed of light $c$. If the dimensionless quantity is written as $e^{\alpha}\varepsilon_0^{\beta}h^{\gamma}c^{\delta}$ and $n$ is a non-zero integer, then $(\alpha,\beta,\gamma,\delta)$ is given by

The dimensions of a cone are measured using a scale with a least count of $2\,\mathrm{mm}$. The diameter of the base and the height are both measured to be $20.0\,\mathrm{cm}$. 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 $10 \pm 0.1$ cm and the distance of its real image from the lens is $20 \pm 0.2$ cm. The error in the determination of focal length of the lens is $n$ %. The value of $n$ is _____.

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 $Y = c^\alpha h^\beta G^\gamma$. 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 $k = \dfrac{1}{4\pi\epsilon_0}$, where $\epsilon_0$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $[B] = [e]^\alpha[m_e]^\beta[h]^\gamma[k]^\delta$. The value of $\alpha + \beta + \gamma + \delta$ is _____.

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.

Measurement condition	Main scale reading	Circular scale reading
Two arms of gauge touching each other without wire	0 division	4 divisions
Attempt-1: With wire	4 divisions	20 divisions
Attempt-2: With wire	4 divisions	16 divisions

What are the diameter and cross-sectional area of the wire measured using the screw gauge?

Identify the pair of physical quantities which have different dimensions :

Identify the pair of physical quantities that have same dimensions :

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

For $z=a^2x^3y^{\frac12}$, 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 ______%.

An expression for a dimensionless quantity P is given by $P=\dfrac\alpha\beta\log_e\left(\dfrac{kt}{\beta x}\right)$; where $\alpha$ and $\beta$ are constants, $x$ is distance; k is Boltzmann constant and t is the temperature. Then the dimensions of $\alpha$ will be :

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 contant will be ______ $\times10^{-2}$ mm.

The dimension of mutual inductance is :

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 ______ $\times10^{-6}$ m.

A silver wire has a mass $(0.6\pm0.006)$ g, radius $(0.5\pm0.005)$ mm and length $(4\pm0.04)$ cm. The maximum percentage error in the measurement of its density will be :

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 $1.5\times10^{11}$ m and its angular diameter is (2000) s when observed from the earth. The diameter of the Sun will be :