Dimensional Analysis

Q. The dimension of coefficient of thermal conductivity is
$(\frac{dQ}{dt}=-KA\frac{dT}{dx})$, where $dQ$ is heat transferred, $dt$ is time taken, $K$ is coefficient of thermal conductivity, $A$ is cross-sectional area of conductor, $dT$ is change in temperature and $dx$ is length of conductor.

1. $\left[M{L}^2{T}^{-2}{K}^{-1}\right]$
2. $\left[ML{T}^{-3}{K}^{-1}\right]$
3. $\left[ML{T}^{-2}{K}^{-1}\right]$
4. $\left[ML{T}^{-3}K\right]$

Q.

The dimensional formula for intensity of magnetising field and magnetic susceptibility are.

1. [M0A2L-1T2]  [M0ALT-1]
2. [M0AL-1T0]  [M0A0L0T0]
3. [MLT-2]  [ML2T-2]
4. [M0A1L-1T2]  [M0A0L0T0]

Q. A system of units uses force $\left(F\right)$, acceleration $\left(A\right)$ and time $\left(T\right)$ as their fundamental physical quantities. The dimension of length in the system is

1. $\left[F^{-1}{T}^2\right]$
2. $\left[F^{-1}{A}^2{T}^{-1}\right]$
3. $\left[F{T}^2\right]$
4. $\left[A{T}^2\right]$

Q. In the equation $\int \frac{dt}{\sqrt{2at-t^2}}=a^x{\sin }^{-1}[\frac{t}{a}-1]$, where '$t$' is time. The value of $x$ is

Q. The period of a body under SHM i.e. presented by T = $P^a{D}^b{S}^c$ ; where P is pressure, D is density and S is surface tension. The value of a, b and c are

1. $-\frac{3}{2},\frac{1}{2}$, 1
2. -1, -2, 3
3. $\frac{1}{2},-\frac{3}{2},-\frac{1}{2}$
4. 1, 2, $\frac{1}{2}$

Q. Young's modulus of a material has the same units as

1. Pressure
2. Strain
3. Compressibility
4. Force

Q. A $100\Omega$ resistance, a $0.1\mu \text{F}$ capacitor and an inductor are connected in series across a $250\text{V}$ supply at variable frequency. Calculate the value of inductance of inductor at which resonance will occur. Given that the resonant frequency is $60\text{Hz}.$

1. $7.03\times {10}^{-5}\text{H}$
2. $0.70\text{H}$
3. $70.4\text{H}$
4. $70.4\text{mH}$

Q. The equation $(P+\frac{a}{v^2})$ (v-b) = constant. If P and V are the pressure and volume, the C.G.S units of a is

1. Dyne $\times c{m}^5$
2. Dyne $\times c{m}^4$
3. $Dyne/c{m}^3$
4. $Dyne/c{m}^2$

Q. If velocity v, acceleration A and force B are chosen as fundamental quantities, then the dimensional formula of angular momentum in terms of v, A and F would be

1. $F{v}^3{A}^{-2}$
2. $F{A}^{-1}v$
3. $F{v}^2{A}^{-1}$
4. $F^2{v}^2{A}^{-1}$

Q. The unit of thermal conductivity is

1. $Jm{K}^{-1}$
2. $J{m}^{-1}{K}^{-1}$
3. $Wm{K}^{-1}$
4. $W{m}^{-1}{K}^{-1}$

Q. Dimensional formula of resolving power of telescope and resolving power of microscope respectively are

1. $M^o{L}^o{T}^o,M^o{L}^o{T}^o$
2. $M^o{L}^o{T}^o,M^o{L}^{-1}{T}^o$
3. $M^o{L}^o{T}^{-1},M^o{L}^{-1}{T}^o$
4. $M^1{L}^1{T}^1,ML{T}^{-2}$

Q. In the ideal gas equation $PV=nRT$, dimensional formula of $R$ is

1. $\left[M{L}^2{T}^{-2}{K}^{-1}mo{l}^{-1}\right]$
2. $\left[M{L}^{2}Kmol\right]$
3. $\left[L^2{T}^{-1}{K}^{-1}mol\right]$
4. $\left[M^2{L}^2{T}^{-2}{K}^{-2}mo{l}^{-1}\right]$

Q. If velocity $V$, acceleration $A$ and force $F$ are taken as fundamental quantities, then the dimensional formula of angular momentum in terms of $V,A$ and $F$ will be

1. $\left[F{A}^{-1}V\right]$
2. $\left[F{A}^{-2}{V}^3\right]$
3. $\left[F{A}^{-1}{V}^2\right]$
4. $\left[F^2{A}^{-1}{V}^2\right]$

Q.

Paragraph for below question

नीचे दिये गये प्रश्न के लिए अनुच्छेद

A series LCR circuit containing a resistance of 100 Ω has angular resonance frequency 2 × 10⁴ rad s^–1. At resonance, the voltages across resistance and inductance are 50 V and 40 V respectively.

एक श्रेणी LCR परिपथ में 100 Ω का एक प्रतिरोध है, इसकी कोणीय अनुनादी आवृत्ति 2 × 10⁴ rad s^–1 है। अनुनाद पर, प्रतिरोध व प्रेरकत्व के सिरों पर वोल्टता क्रमशः 50 V तथा 40 V हैं।

Q. The value of inductance L is

प्रश्न - प्रेरकत्व L का मान है

1. 2 mH
2. 4 mH
3. $\frac{5}{3}\text{mH}$
4. 5 mH

Q. The equation of the stationary wave is
$y=2A\sin \left(\frac{2\pi ct}{\lambda }\right)\cos \left(\frac{2\pi x}{\lambda }\right)$
Which of the following statements is wrong?

1. The unit of $ct$ is same as that of $\lambda$
2. The unit of $x$ is same as that of $\lambda$
3. The unit of $\frac{2\pi c}{\lambda }$ is same as that of $\frac{2\pi x}{\lambda t}$
4. The unit of $\frac{c}{\lambda }$ is same as that of $\frac{x}{\lambda }$

Q. A physical quantity $x$ depends on other physical quantities $y$ and $z$ as $x=Ay+B\tan \left(Cz\right)$, where $A,B$ and $C$ are constants. Which of the following do not have the same dimension?

1. $x$ and $B$
2. $C$ and $z^{-1}$
3. $y$ and $B/A$
4. $x$ and $A$

Q. If we take units of mass, length and time as $2\text{kg},2\text{m}$ and $2\text{s}$ respectively then the unit of energy will be

1. $8\text{J}$
2. $6\text{J}$
3. $4\text{J}$
4. $2\text{J}$

Q. A spherical liquid drop is placed on a horizontal plane. A small disturbance causes the volume of the drop to oscillate. The time period of oscillation $\left(T\right)$ of the liquid drop depends on radius $\left(r\right)$ of the drop, density $\left(\rho \right)$ and surface tension $\left(S\right)$ of the liquid. Which among the following will be a possible expression for $T$ (where $k$ is a dimensionless constant)? (given surface tension is force per unit length)

1. $k\sqrt{\frac{\rho r}{S}}$
2. $k\sqrt{\frac{{\rho }^{2}r}{S}}$
3. $k\sqrt{\frac{\rho r^3}{S}}$
4. $k\sqrt{\frac{\rho r^3}{S^2}}$

Q.

This section contains 1 Assertion-Reason type question, which has 4 choices (a), (b), (c) and (d) out of which ONLY ONE is correct.
इस खण्ड में 1 कथन-कारण प्रकार का प्रश्न है, जिसमें 4 विकल्प (a), (b), (c) तथा (d) दिये गये हैं, जिनमें से केवल एक सही है।

A : Dimensional formulas of work, energy and torque are same.
A : कार्य, ऊर्जा तथा बल आघूर्ण के विमीय सूत्र समान होते हैं।

R : Two or more than two quantities can have same dimensional formula but the converse is not true.
R : दो या दो से अधिक राशियों का विमीय सूत्र समान हो सकता है लेकिन इसका विलोम सत्य नहीं है।

1. Both (A) and (R) are true and (R) is the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं तथा (R), (A) का सही स्पष्टीकरण है
2. Both (A) and (R) are true but (R) is not the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं लेकिन (R), (A) का सही स्पष्टीकरण नहीं है
3. (A) is true but (R) is false
   
   A) सही है लेकिन (R) गलत है
4. (A) is false but (R) is true
   
   (A) गलत है लेकिन (R) सही है

Q. The dimensions of the velocity and acceleration are same.

1. False
2. True

Q. E, m, l and G denote energy, mass, angular momentum and gravitational constant respectively, then the dimension of $\frac{E{l}^2}{m^2{G}^2}$ are

1. $\left[M^0{L}^0{T}^0\right]$
2. $\left[M^0{L}^0{T}^1\right]$
3. $\left[M^0{L}^1{T}^1\right]$
4. $\left[M^0{L}^1{T}^{-1}\right]$

Q. Frequency is the function of density $\left(\rho \right)$, length $\left(a\right)$ and surface tension $\left(T\right)$. Then its value is ($k$ is a constant)

1. $k{\rho }^{-1/2}{a}^{-3/2}{T}^{1/2}$
2. $k{\rho }^{1/2}{a}^{3/2}{T}^{-1/2}$
3. $k{\rho }^{1/2}{a}^{-3/2}{T}^{1/2}$
4. $k{\rho }^{-1/2}{a}^{3/2}{T}^{-1/2}$

Q. The mass of liquid flowing per second per unit area of cross section of the tube is proportional to $P^x$ and $v^y$, where $P$ is the pressure difference and $v$ is the velocity, then the relation between $x$ and $y$ is

1. $x=y$
2. $x=-y$
3. $y2=x$
4. $y=-x^2$

Q. Density of wood is $0.5\text{g/cc}$ in the $CGS$ system of units. The numerical value of density in $MKS$ system of units is

1. $800$
2. $500$
3. $600$
4. $700$

Q. If velocity is given as $v=\alpha -\beta x$, then dimensional formula of $\beta {\alpha }^2$ is

1. $\left[M^{0}L{T}^{-2}\right]$
2. $\left[M^1{L}^2{T}^{-2}\right]$
3. $\left[M^0{L}^2{T}^{-3}\right]$
4. $\left[M^0{L}^3{T}^{-3}\right]$

Q. Terminal velocity of Mars rover depends upon
P = gas density of atmosphere of mars
A = frontal area
W = weight of falling object
What may be the possible relation between terminal velocity $v$ and given quantity.

1. $v=\frac{w}{pA}$
2. $v=\sqrt{\frac{pA}{w}}$
3. $v=\sqrt{\frac{p{A}^3}{w}}$
4. $v=\sqrt{\frac{w}{pA}}$

Q.

Find the expression for the time period of a pendulum using dimensional analysis.

1. $k\sqrt{\frac{g}{l}}$

2. $k\sqrt{\frac{l}{g}}$

3. kl

4. $\frac{kl}{g}$

Q. In S = a + bt + $c{t}^2$. S is measured in metres and t in seconds. The unit of c is

1. No dimensions
2. m
3. $m{s}^{-1}$
4. $m{s}^{-2}$

Q.

Dimensions of $\frac{1}{{\mu }_0{ϵ}_0}$, where symbols have their usual meaning, are

1. $\left[L{T}^{-1}\right]$

2. $\left[L^{-1}T\right]$

3. $\left[L^{-2}{T}^2\right]$

4. $\left[L^2{T}^{-2}\right]$

Q. The frequency of vibration f of a mass m suspended from a spring of spring constant K is given by a relation of this type f =C$m^x{K}^y$; where C is a dimensionless quantity. The value of x and y are

1. x = $\frac{1}{2}$, y = $\frac{1}{2}$
2. x = -$\frac{1}{2}$, y = - $\frac{1}{2}$
3. x = $\frac{1}{2}$, y = - $\frac{1}{2}$
4. x = - $\frac{1}{2}$, y = $\frac{1}{2}$