Unit And Dimensions Set-2

Q1: If $E$ and $H$ represents the intensity of electric field and magnetising field respectively, then the unit of $E/H$ will be:

Q2: 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 which of the following quantity (ies):

Q3: The density of a material in SI units is 128 kg m$^{-3}$. In certain units in which the unit of length is 25 cm and the unit of mass 50 g, the numerical value of density of the material is

Q4: The dimensional formula of latent heat is:

Q5: If $\epsilon_0$ is the permittivity of free space and $E$ is the electric field, then $\epsilon_0 E^2$ has the dimensions:

Q6: Given below are two statements:
Statement (I): Dimensions of specific heat is $[L T^{-2} K^{-1}]$
Statement (II): Dimensions of gas constant is $[M L T^{-1} K^{-1}]$

Q7: What is the dimensional formula of $a b^{-1}$ in the equation $\left( p + \frac{a}{V^2} \right) (V - b) = RT$, where letters have their usual meaning.

Q8: If $G$ be the gravitational constant and $u$ be the energy density then which of the following quantity has the dimension as that of $\sqrt{u G}$:

Q9: The dimensional formula of angular impulse is:

Q10: Given below are two statements:
Statement-I: Planck's constant and angular momentum have same dimensions.
Statement-II: Linear momentum and moment of force have same dimensions.

Q11: Applying the principle of homogeneity of dimension, which one is correct? where $T$ is time period, $G$ is gravitational constant, $M$ is mass, $r$ is radius of orbit.

Q12: A force is represented by $F = a x^2 + b t^{1/2}$ where $x$ = distance and $t$ = time. The dimensions of $b^2 / a$ are:

Q13: If mass is written as $m = k c^P G^{-1/2} h^{1/2}$ then the value $P$ will be: (Constants have their usual meaning with $k$ dimensionless constant)

Q14: The equation of state of a real gas is given by $\left(P + \frac{a}{V^2}\right)(V - b) = RT$ where $P$, $V$ and $T$ are pressure volume and temperature respectively and $R$ is universal gas constant. The dimensions of $\frac{a}{b^2}$ is similar to that of:

Q15: The frequency $\nu$ of an oscillating liquid drop may depend upon radius $r$ of the drop, density $\rho$ of liquid and the surface tension $s$ of the liquid as: $\nu = r^a \rho^b s^c$. The values of $a, b$ and $c$ respectively are

Q16: If time (t), velocity (v), and angular momentum (l) are taken as the fundamental units. Then the dimension of mass (m) in terms of t, v and l is:

Q17: In a typical combustion engine the work done by a gas molecule is given by $W = \alpha^2 \beta e^{-\frac{\alpha x^2}{k T}}$, 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:

Q18: Dimensional formula for thermal conductivity is (here $K$ denotes the temperature):

Q19: Expression for time in terms of $G$ (universal gravitational constant), $h$ (Planck constant) and $c$ (speed of light) is proportional to:

Q20: Young's modulus is determined by the equation given by $Y = 49000 \frac{m}{\ell}$ where $m$ is the mass and $\ell$ is the extension of wire used in the experiment. Now error in Young modulus (Y) is estimated by taking data from $m-\ell$ plot in graph paper. The smallest scale divisions are 5 g and 0.02 cm along load axis and extension axis respectively. If the value of $m$ and $\ell$ are 500 g and 2 cm respectively then percentage error of $Y$ is:

Q21: To find the spring constant ($k$) of a spring experimentally, a student commits 2% positive error in the measurement of time and 1% negative error in measurement of mass. The percentage error in determining value of $k$ is:

Q22: Time periods of oscillation of the same simple pendulum measured using four different measuring clocks were recorded as 4.62 s, 4.632 s, 4.6 s and 4.64 s. The arithmetic mean of these reading in correct significant figure is:

Q23: In an experiment to measure focal length ($f$) of convex lens, the least counts of the measuring scales for the position of object ($u$) and for the position of image ($v$) are $\Delta u$ and $\Delta v$, respectively. The error in the measurement of the focal length of the convex lens will be:

Q24: The radius $r$, length $l$ and resistance $R$ of a metal wire was measured in the laboratory as $r = (0.35 \pm 0.05)$ cm, $R = (100 \pm 10)$ ohm, $l = (15 \pm 0.2)$ cm. The percentage error in resistivity of the material of the wire is:

Q25: The measured value of the length of a simple pendulum is 20 cm with 2 mm accuracy. The time for 50 oscillations was measured to be 40 seconds with 1 second resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is $N\%$. The value of $N$ is:

Q26: If the percentage errors in measuring the length and the diameter of a wire are 0.1% each. The percentage error in measuring its resistance will be:

Q27: The resistance $R = \frac{V}{I}$ where $V = (200 \pm 5)V$ and $I = (20 \pm 0.2)A$, the percentage error in the measurement of $R$ is:

Q28: A physical quantity $Q$ is found to depend on quantities $a, b, c$ by the relation $Q = \frac{a^4 b^3}{c^2}$. The percentage error in $a, b$ and $c$ are 3%, 4% and 5% respectively. Then, the percentage error in $Q$ is:

Q29: Two resistances are given as $R_1 = (10 \pm 0.5) \Omega$ and $R_2 = (15 \pm 0.5) \Omega$. The percentage error in the measurement of equivalent resistance when they are connected in parallel is

Q30: If $Z = \frac{A^2 B^3}{C^4}$, then the relative error in $Z$ will be: