Applications of Dimensional Analysis

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

Where $H =$ magnetic field intensity; $E =$ electric field, $\epsilon =$ permittivity, $x =$ distance, $t =$ time The values of $p, q, r$ and $s$ respectively are:

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 constants with appropriate dimensions. The dimensions of $AB$ are ______

$L, C$ and $R$ represents physical quantities inductance, capacitance and resistance respectively. The dimensional formula $\mathrm{ML^{2}T^{-4}A^{-2}}$ corresponds to ______.

The dimensional formula of $\frac{1}{2}\epsilon_0 E^2$ ($\epsilon_0$ = permittivity of vacuum and $E$ = electric field) is $M^aL^bT^c$. The value of $2a - b + c =$ ________.

Consider a modified Bernoulli equation.

$\left(\mathrm{P} + \frac{A}{Bt^2}\right) + \rho g(h + Bt) + \frac{1}{2}\rho V^2 =$ constant

If $t$ has the dimension of time then the dimensions of $A$ and $B$ are ______ respectively.

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

The speed of light in vacuum is taken as unity. If light takes 6 min 40 s to reach the Earth from the Sun, the distance between the Sun and the Earth in new unit 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?

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 $\frac{b+c}{a+d}$ is ______.

Match List - I with List - II.

where h (Planck's constant), G (gravitational constant) and c (speed of light in vacuum) as

fundamental units.

Choose the correct answer from the options given below:

In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of $M^PL^QT^RA^S$, where value of ' $Q$ ' and ' $R$ ' are

The equation for real gas is given by $\left(\mathrm{P} + \frac{\mathrm{a}}{\mathrm{V}^2}\right)(\mathrm{V} - \mathrm{b}) = \mathrm{RT}$, where $\mathrm{P, V, T}$ and R are the pressure, volume, temperature and gas constant, respectively. The dimension of $\mathrm{ab^{-2}}$ is equivalent to that of:

The expression given below shows the variation of velocity (v) with time (t),

$v = \mathrm{At^{2}} + \frac{\mathrm{Bt}}{\mathrm{C+t}}$.

The dimension of ABC is:

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

A balloon is made of a material of surface tension $S$ and its inflation outlet (from where gas is filled in it) has small area $A$. It is filled with a gas of density $\rho$ and takes a spherical shape of radius $R$. When the gas is allowed to flow freely out of it, its radius $r$ changes from $R$ to 0 (zero) in time $T$. If the speed $v(r)$ of gas coming out of the balloon depends on $r$ as $r^a$ and $T \propto S^\alpha A^\beta \rho^\gamma R^\delta$ then

The potential energy of a particle moving along $x$-direction varies as $V = \frac{Ax^2}{\sqrt{x} + B}$. The dimensions of $\frac{A^2}{B}$ are:

A force defined by $F = \alpha t^2 + \beta t$ acts on a particle at a given time $t$. The factor which is dimensionless, if $\alpha$ and $\beta$ are constants, is:

The de-Broglie wavelength associated with a particle of mass $m$ and energy $E$ is $h/\sqrt{2mE}$. The dimensional formula for Planck's constant is:

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

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

The equation of stationary wave is:

$y = 2\mathrm{a}\sin\left(\frac{2\pi\mathrm{nt}}{\lambda}\right)\cos\left(\frac{2\pi x}{\lambda}\right)$.

Which of the following is NOT correct:

Consider two physical quantities $A$ and $B$ related to each other as $E = \frac{B - x^2}{At}$ where $E, x$ and $t$ have dimensions of energy, length and time respectively. The dimension of $AB$ is

A force is represented by $F = ax^2 + bt^{\frac{1}{2}}$

where $x =$ distance and $t =$ time. The dimensions of $b^2/a$ are:

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

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

In the equation $\left[X + \frac{a}{Y^2}\right][Y - b] = \mathrm{R}T, X$ is pressure, $Y$ is volume, $\mathrm{R}$ is universal gas constant and $T$ is temperature. The physical quantity equivalent to the ratio $\frac{a}{b}$ is:

If force (F), velocity (V) and time (T) are considered as fundamental physical quantities, then dimensional formula of density will be:

If the velocity of light $\mathrm{c}$, universal gravitational constant $\mathrm{G}$ and Planck's constant $\mathrm{h}$ are chosen as fundamental quantities. The dimensions of mass in the new system is:

$\left(P + \frac{a}{V^2}\right)(V - b) = RT$ represents the equation of state of some gases. Where $P$ is the pressure, $V$ is the volume, $T$ is the temperature and $a, b, R$ are the constants. The physical quantity, which has dimensional formula as that of $\frac{b^2}{a}$, will be:

The equation of a circle is given by $x^2 + y^2 = a^2$, where a is the radius. If the equation is modified to change the origin other than (0, 0), then find out the correct dimensions of A and B in a new equation: $(x - At)^2 + \left(y - \frac{t}{B}\right)^2 = a^2$. The dimensions of t is given as $[\mathrm{T^{-1}}]$.

Match List I with List II:

Choose the correct answer from the options given below:

The speed of a wave produced in water is given by $v = \lambda^{a}g^{b}\rho^{c}$. Where $\lambda, g$ and $\rho$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of $a, b$ and $c$ respectively, are:

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

In van der Waal 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:

If current (I), kinetic energy (K) and charge (Q) are taken as the fundamental quantities, The dimensional representation of power will be:

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

Consider the efficiency of Carnot's engine is given by $\eta = \frac{\alpha\beta}{\sin\theta}\log_e\frac{\beta x}{kT}$, where $\alpha$ and $\beta$ are constants. If T is temperature, k is Boltzmann constant, $\theta$ is angular displacement and x has the dimensions of length. Then, choose the incorrect option:

Velocity (v) and acceleration (a) in two systems of units 1 and 2 are related as $v_2 = \frac{n}{m^2}v_1$ and $a_2 = \frac{a_1}{mn}$ respectively. Here m and n are constants. The relations for distance and time in two systems respectively are:

An expression for a dimensionless quantity P is given by $P = \frac{\alpha}{\beta}\log_e\left(\frac{kt}{\beta x}\right)$; 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:

The force is given in terms of time t and displacement x by the equation

F = A cos Bx + C sin Dt

The dimensional formula of $\frac{AD}{B}$ is:

If force [F], acceleration [A] and time [T] are chosen as the fundamental physical quantities. Find the dimensions of energy.

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

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

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:

If E, L, 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 = EL2M$-5\mathrm{G}-$2 are:

The entropy of any system is given by

$S = \alpha^2\beta\ln\left[\frac{\mu kR}{J\beta^2} + 3\right]$ where $\alpha$ and $\beta$ are the constants. $\mu$, J, k and R are no. of moles, mechanical equivalent of heat, Boltzmann constant and gas constant respectively.

[Take $S = \frac{dQ}{T}$ ]

Choose the incorrect option from the following:

In a typical combustion engine the workdone by a gas molecule is given by $W = \alpha^2\beta e^{\frac{-\beta x^2}{kT}}$, 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:

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 be:

Which of the following equations is dimensionally incorrect?

Where t = time, h = height, s = surface tension, $\theta$ = angle, $\rho$ = density, a, r = radius, g = acceleration due to gravity, v = volume, p = pressure, W = work done, $\Gamma$ = torque, $\epsilon$ = permittivity, E = electric field, J = current density, L = length.

Let force $F = A\sin(Ct) + B\cos(Dx)$, where $x$ and $t$ are displacement and time, respectively. The dimensions of $\frac{C}{D}$ are same as dimensions of

If momentum (P), area (A) and time (T) are taken to be the fundamental quantities then the dimensional formula for energy is

The dimension of stopping potential V0 in photoelectric effect in units of Planck's constant 'h', speed of light 'c' and Gravitational constant 'G' and ampere A is:

A quantity x is given by $\left(\frac{IFv^2}{WL^4}\right)$ in terms of moment of inertia I, force F, velocity v, work W and Length L. The dimensional formula for x is same as that of:

If speed V, area A and force F are chosen as fundamental units, then the dimension of Young’s modulus will be

A quantity f is given by $f = \sqrt{\frac{hc^5}{G}}$ where c is speed of light, G universal gravitational constant and h is the Planck's constant. Dimension of f is that of:

The force ' $F$ ' acting on a body of density ' $d$ ' are related by the relation $F = \frac{y}{\sqrt{d}}$. The dimensions of ' $y$ ' are

In the formula X = 5YZ2, X and Z have dimensions of capacitance and magnetic field, respectively. What are the dimensions of Y in SI units?

If speed (V), acceleration (A) and force (F) are considered as fundamental units, the dimension of Young's modulus will be:

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

The force of interaction between two atoms is given by F = $\alpha\beta$exp $\left(-\frac{x^2}{\alpha kT}\right)$; where x is the distance, k is the Boltzmann constant and T is temperature and $\alpha$ and $\beta$ are two constants. The dimension of $\beta$ is:

If surface tension (S), Moment of inertia (I) and Planck's constant (h), were to be taken as the fundamental units, the dimensional formula for linear momentum would be :-

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 is 50 g, the numerical value of density of the material is -

The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be determined from a suitable combination of the fundamental physical constants G, h and c. Which of the following correctly gives the Planck length?

Time (T), velocity (C) and angular momentum (h) are chosen as fundamentalquantities instead of mass, length and time. In terms of these, the dimensions of mass would be:

A physical quantity of the dimensions of length that can be formed out of c, G and $\frac{e^2}{4\pi\varepsilon_0}$ is [c is velocity of light, G is the universal constant of gravitation and e is charge]

A, B, C and D are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation AD = C ln (BD) holds true. Then which of the combination is not a meaningful quantity?

Planck's constant (h), speed of light in vacuum (c) and Newton's gravitional constant (G) are three fundamental constants. Which of the following combinations of these has the dimension of length?

If energy (E), velocity (V) and time (T) are chosen as the fundamental quantities, the dimensional formula of surface tension will be

If electronic charge e, electron mass m, speed of light in vacuum c and Planck's constant h are taken as fundamental quantities, the permeability of vacuum $\mu$0 can be expressed in units of: