If \( \varepsilon_0 \) is the permittivity of free space and \( E \) is the electric field, then \( \varepsilon_0 \) \(E^{2} \) has the dimensions:
\( [\, M L^{2} T^{-2} \,] \)
\( [\, M^{0} L^{-2} TA \,] \)
\( [\, M L^{-1} T^{-2} \,] \)
\( [\, M^{-1} L^{-3} T^{4} A^{2} \,] \)
The magnitudes of \( x \) and \( y \) components of \( \vec{P} \) are \( 5 \) and \( 3 \) respectively. The magnitudes of \( x \) and \( y \) components of \( \vec{P}-\vec{Q} \) are \( 9 \) and \( 7 \), respectively. Then the magnitude of \( \vec{Q} \) is:
Match List I with List II.
Choose the correct answer from the options given below:
In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system.
The measured quantity has dimension of $[M^a L^b T^c]$. If $b = 3$, the value of $c$ is ________.
What is the dimensional formula of \( a b^{-1} \) in the equation \( \left( P + \dfrac{a}{V^{2}} \right) ( V - b ) = R T \), where letters have their usual meaning.
The dimensions of \( \left( \dfrac{B^{2}}{\mu_{0}} \right) \) will be: (if \( \mu_{0} \) : permeability of free space and \( B \) : magnetic field)
Consider a modified Bernoulli equation.\( \left( P + \frac{A}{B t^{2}} \right) + \rho g (h + B t) + \frac{1}{2} \rho v^{2} = \text{constant}. \) If \( t \) has the dimension of time then the dimensions of \( A \) and \( B \) are ______ , ______ respectively.
If force \( (F) \), velocity \( (V) \) and time \( (T) \) are considered as fundamental physical quantity, then dimensional formula of density will be:
Identify the pair of physical quantities that have same dimensions:
Given below are two statements: Statement (I) : Dimensions of specific heat is \( [ L^{2} T^{-2} K^{-1} ] \) Statement (II) : Dimensions of gas constant is \( [ M L^{2} T^{-1} K^{-1} ] \) In the light of the above statements, choose the most appropriate answer from the options given below.
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: \( (x-At)^{2}+\left(y-\dfrac{t}{B}\right)^{2}=a^{2} \). The dimension of \( t \) \( [T^{-1}] \).
Electric field in a certain region is given by \( \vec{E} = \left( \frac{A}{x^{2}} \hat{i} + \frac{B}{y^{3}} \hat{j} \right) \). The SI unit of \( A \) and \( B \) are:
Given below are two statements: One is labelled as Assertion (A) and other is labelled as Reason (R).Assertion (A): Time period of oscillation of a liquid drop depends on surface tension \( (S) \), if density of the liquid is \( \rho \) and radius of the drop is \( r \), then \( T = K \sqrt{\dfrac{\rho r^{3}}{S^\dfrac{3}{2}}} \) is dimensionally correct, where \( K \) is dimensionless.Reason (R): Using dimensional analysis we get R.H.S. having different dimension than that of time period.In the light of above statements, choose the correct answer from the options given below.
Given a charge \( q \), current \( I \) and permeability of vacuum \( \mu_{0} \). Which of the following quantity has the dimension of momentum?
If velocity \( [V] \) time \( [T] \) and force \( [F] \) are chosen as the base quantities, the dimensions of the mass will be:
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.In the light of the above statements, choose the correct answer from the options given below :
A force is represented by \( F = a x^{2} + b t^{1/2} \) where \( x \) = distance and \( t \) = time. The dimensions of \( \frac{b^{2}}{a} \) are:
Least count of a vernier caliper is \(\displaystyle \frac{1}{20N} \) cm. The value of one division on the main scale is \( 1 \, \text{mm} \). Then the number of divisions of main scale that coincide with \( N \) division of vernier scale is:
An electron having de-Broglie wavelength \( \lambda \) is incident on a target in a X-ray tube. Cut-off wavelength of emitted X-ray is:
Match List-I with List-II
Consider two physical quantities \( A \) and \( B \) related to each other as \( E = \dfrac{B - x^2}{A t} \) where \( E, x \) and \( t \) have dimensions of energy, length and time respectively. The dimension of \( AB \) is:
The energy of a system is given as \( E(t) = \alpha^{3} e^{-\beta t} \), where \( t \) is the time and \( \beta = 0.3 \, \text{s}^{-1} \). The errors in the measurement of \( \alpha \) and \( t \) are \( 1.2\% \) and \( 1.6\% \), respectively. At \( t = 5 \, \text{s} \), maximum percentage error in the energy is:
An expression of energy density is given by \( u = \dfrac{\alpha}{\beta}\sin\!\left(\dfrac{\alpha x}{k t}\right) \), where \( \alpha, \beta \) are constants, \( x \) is displacement, \( k \) is Boltzmann constant and \( t \) is the temperature. The dimensions of \( \beta \) will be:
The velocity \( (v) \) of a body is varying with time \( t \) as given by the equation \( v=a+\dfrac{b}{t}+\dfrac{c}{t^{2}} \). Based on above information, match the following.
A body of mass \( (5 \pm 0.5) \, \text{kg} \) is moving with a velocity of \( (20 \pm 0.4) \, \text{m/s} \). Its kinetic energy will be:
In an electromagnetic system, a quantity defined as the ratio of electric dipole moment and magnetic dipole moment has dimension of \( M^{P} L^{Q} T^{R} A^{S} \). The value of \( P \) and \( Q \) are:
A particle is moving on a circular path with constant speed \( v \). The magnitude of the change in its velocity after it has described an angle of \( 120^\circ \) is:
Match List I with List IIChoose the correct answer from the options given below:
If force \( (F) \), length \( (L) \) and time \( (T) \) are taken as the fundamental quantities. Then what will be the dimension of density:
The potential energy of a particle changes with distance \( x \) from a fixed origin as \( V=\dfrac{A\sqrt{x}}{x+B} \), where \( A \) and \( B \) are constant with appropriate dimensions. The dimensions of \( AB \) are ______.
Match List-I with List-IIChoose the most appropriate answer from the option given below:
Given below are two statements: One is labelled a Assertion A and the other is labelled as Reason R.
Assertion A: Product of Pressure ($P$) and time ($t$) has the same dimension at that of coefficient of viscosity.Reason R: $\text{Coefficient of viscosity} = \frac{\text{Force}}{\text{Velocity gradient}}$.
The angle which a vector \( \hat{i}-\hat{j}+\sqrt{2}\,\hat{k} \) makes with y-axis is:
If the velocity of light \( c \), universal gravitational constant \( G \) and Planck’s constant \( h \) are chosen as fundamental quantities. The dimensions of mass in the new system is:
The de-Broglie wavelength associated with a particle of mass \(m\) and energy \(E\) is \( \dfrac{h}{\sqrt{2 m E}} \). The dimensional formula for Planck's constant is:
Consider a physical quantity \( (Y) \) defined as \( Y=\dfrac{F^{2}}{m v L} \) where \( F, m, v, L \) represents force, mass, velocity and length respectively. Difference of dimension of mass and time in \( Y \) is ________.
Match List - I with List - II:
Choose the most appropriate answer from the options given below :
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.
Match the List-I with List-II.
Which of the following is not a dimensionless quantity?
Which one of the following is the correct dimensional formula for the capacitance in \( F \)? \( M \), \( L \), \( T \) and \( C \) stand for unit of mass, length, time and charge,
For the determination of refractive index of glass slab, a travelling microscope is used whose main scale contains \( 300 \) equal divisions equals to \( 15 \,\text{cm} \). The vernier scale attached to the microscope has \( 25 \) divisions equals to \( 24 \) divisions of main scale. The least count (\( LC \)) of the travelling microscope is (in \( \text{cm} \)):
Match List-I with List-IIChoose the correct answer from the options given below:
The speed of a wave produced in water is given by \( \nu = \lambda^{a} g^{b} \rho^{c} \). 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:
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 \( 6 \,\text{min}\, 40 \,\text{s} \) to cover this distance?
If \( \varepsilon E \) and \( t \) represent the free space permittivity, electric field and time respectively, then the unit of \( \dfrac{\varepsilon E}{t} \) will be:
The SI unit of a physical quantity is pascal-second. The dimensional formula of this quantity will be:
Match List I and List II :
Choose the correct answer from the options given below :
The dimensional formula of latent heat is:
Match List - I with List - II.Choose the correct answer from the options given below:
Match List-I with List-II.
Dimensions of universal gravitational constant \( (G) \) in terms of Planck’s constant \( (h) \), distance \( (L) \), mass \( (M) \) and time \( (T) \) are ________.
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 \( P = E L^{2} M^{-5} G^{-2} \) are:
Which of the following physical quantities have the same dimensions?
Match List I with List II:Choose the correct answer from the options given below:
The dimension of mutual inductance is:
If momentum \( [P] \), area \( [A] \) and time \( [T] \) are taken as fundamental quantities, then the dimensional formula for coefficient of viscosity is :
\( L \), \( C \) and \( R \) represents physical quantities inductance, capacitance and resistance respectively. The dimensional formula \( M L^{2} \, T^{-3} \, A^{-2} \) corresponds to ______.
The angle between vector \( \vec{Q} \) and the resultant of \( (2\vec{Q} + 2\vec{P}) \) and \( (2\vec{Q} - 2\vec{P}) \) is:
The equation of state of real gas is given by \(\displaystyle \left( p + \frac{a}{V^{2}} \right)(V-b)=RT \), where \( P \), \( V \) and \( T \) are pressure, volume and temperature respectively and \( R \) is the universal gas constant. The dimensions of \(\displaystyle \frac{a}{b^{2}} \) is similar to that of:
If \( \mu_{0} \) and \( \epsilon_{0} \) are the permeability and permittivity of free space, respectively, then the dimension of \( \left( \frac{1}{\mu_{0}\epsilon_{0}} \right) \) is:
Three vectors \( \overrightarrow{OP} \), \( \overrightarrow{OQ} \) and \( \overrightarrow{OR} \) each of magnitude \( A \) are acting as shown in figure. The resultant of the three vectors is \( A\sqrt{x} \). The value of \( x \) is _______.
For \( z = a^{2} x^{3} y^{\frac{1}{2}} \), 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 ____________ \( \% \).
For three vectors \( \vec{A} = (-x\hat{i} - 6\hat{j} - 2\hat{k}) \), \( \vec{B} = (-\hat{i} + 4\hat{j} + 3\hat{k}) \) and \( \vec{C} = (-8\hat{i} - \hat{j} + 3\hat{k}) \), if \( \vec{A}\cdot(\vec{B}\times\vec{C}) = 0 \), then value of \( x \) is ________.
Dimension of \( \dfrac{1}{\mu_0 \, \epsilon_0} \) should be equal to:
Identify the pair of physical quantities which have different dimensions:
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:
A physical quantity \( P \) is given as \( P = \dfrac{a^{2} b^{3}}{c \sqrt{d}} \)The percentage error in the measurement of \( a \), \( b \), \( c \) and \( d \) are \( 1\% \), \( 2\% \), \( 3\% \) and \( 4\% \) respectively. The percentage error in the measurement of quantity \( P \) will be:
The percentage error in the calculated volume of a sphere, if there is \( 2\% \) error in its diameter measurement, is_________.
Match Column I with Column II
Choose the correct answer form the options given below:
There are \( 100 \) divisions on the circular scale of a screw gauge of pitch \( 1 \, \text{mm} \). With no measuring quantity in between the jaws, the zero of the circular scale lies \( 5 \) divisions below the reference line. The diameter of a wire is then measured using this screw gauge. It is found that \( 4 \) linear scale divisions are clearly visible while \( 60 \) divisions on circular scale coincide with the reference line. The diameter of the wire is:
The dimensional formula of angular impulse is:
If \( R \), \( X_L \), and \( X_C \) represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless:
The dimensional formula of \( \frac{1}{2}\varepsilon_{0}E^{2} \) (\( \varepsilon_{0} = \) permittivity of vacuum and \( E = \) electric field) is \( M^{a}L^{b}T^{c} \). The value of \( 2a - b + c \) is ______.
The expression given below shows the variation of velocity \( (v) \) with time \( (t) \), \( v = A t^2 + \frac{B t}{C + t} \). The dimension of \( ABC \) is:
The equation of stationary wave is: \( y = 2a \sin\!\left( \dfrac{2\pi n t}{\lambda} \right)\cos\!\left( \dfrac{2\pi x}{\lambda} \right) \). Which of the following is NOT correct:
The distance of the Sun from earth is \( 1.5 \times 10^{11} \, \text{m} \) and its angular diameter is \( (2000) \, s \) when observed from the earth. The diameter of the Sun will be:
If \( L \), \( C \) and \( R \) are the self inductance, capacitance and resistance respectively, which of the following does not have the dimension of time?
An expression for a dimensionless quantity \( P \) is given by \( P = \frac{\alpha}{\beta} \log_e \left( \frac{k t}{\beta x} \right) \); 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:
If \( B \) is magnetic field and \( \mu_0 \) is permeability of free space, then the dimensions of \( (B / \mu_0) \) is:
The entropy of any system is given by \( S = \alpha^2 \beta \ln \left[ \dfrac{\mu k R}{J \beta^2} + 3 \right] \) 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 \( S = \dfrac{dQ}{T} \)]Choose the incorrect option from the following:
If the Kinetic energy of a moving body becomes four times its initial Kinetic energy, then the percentage change in its momentum will be:
In an experiment of measuring the refractive index of a glass slab using travelling microscope in physics lab, a student measures real thickness of the glass slab as \( 5.25 \, \text{mm} \) and apparent thickness of the glass slab as \( 5.00 \, \text{mm} \). Travelling microscope has \( 20 \) divisions in one \( \text{cm} \) on main scale and \( 50 \) divisions on vernier scale is equal to \( 49 \) divisions on main scale. The estimated uncertainty in the measurement of refractive index of the slab is \( \dfrac{x}{10} \times 10^{-3} \), where \( x \) is __________.
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:
If mass is written as \( m = k c^{P} G^{-1/2} h^{1/2} \) then the value of \( P \) will be: (Constants have their usual meaning with \( k \) a dimensionless constant).
Match List-I with List-II.Choose the most appropriate answer from the options given below:
If \( G \) be the gravitational constant and \( u \) be the energy density then which of the following quantity have the dimensions as that of the \( \sqrt{uG} \):
The measured value of the length of a simple pendulum is \( 20 \, \text{cm} \) with \( 2 \, \text{mm} \) accuracy. The time for \( 50 \) oscillations was measured to be \( 40 \, \text{seconds} \) with \( 1 \, \text{seconds} \) resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is \( N\% \). The value of \( N \) is:
The pair of physical quantities not having same dimensions is:
Match List I with List II:
Match List I with List II
In an expression \( a \times 10^{b} \).
Consider aa physical quantity (Y) defined as\( Y=\dfrac{F^2}{m v L} \)where \( F, m, v, L \) represents force, mass, velocity and length respectively. Difference of dimension of mass and time in \( Y \) is ________.
The electric flux is \( \phi = \alpha \sigma + \beta \lambda \), where \( \lambda \) and \( \sigma \) are linear and surface charge density, respectively. \( \left( \frac{\alpha}{\beta} \right) \) represents:
A person measures mass of \( 3 \) different particles as \( 435.42 \, \text{g} \), \( 226.3 \, \text{g} \) and \( 0.125 \, \text{g} \). According to the rules for arithmetic operations with significant figures, the addition of the mass of \( 3 \) particles will be:
Match the LIST-I with LIST-II
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 _______.
Keeping the significant figures in view, the sum of the physical quantities \( 52.01 \, \text{m} \), \( 153.2 \, \text{m} \) and \( 0.123 \, \text{m} \) is:
Match List – I with List – II
Choose the correct answer from the options given below ;
