The potential energy of a particle varies the distance x from a fixed origin as U = \dfrac{A \sqrt{x}}{x^2 + B} , where A and B are dimensional constants, then find the dimensional formula for AB.

x= distance from a fixed origin u=\dfrac { A\sqrt { x } }{ { x }^{ 2 }+B } unit of B is same as { x }^{ 2 } . Unit of { x }^{ 2 }=\left[ { L }^{ 2 } \right] B=\left[ { L }^{ 2 } \right] u=\dfrac { \left[ { x }^{ 1/2 } \right] A }{ { x }^{ 2 }+B } A=\dfrac { u\left( { x }^{ 2 }+B \right) }{ \sqrt { x } } =\dfrac { { kgm }^{ 2 } }{ { s }^{ 2 } } \times \dfrac { { m }^{ 2 } }{ { m }^{ 1/2 } } A=\dfrac { kg{ m }^{ 7/2 } }{ { s }^{ 2 } } A=\left[ { ML }^{ 7/2 }{ T }^{ -2 } \right] Dimensions of AB=\left[ { ML }^{ 7/2 }{ T }^{ -2 } \right] \left[ { L }^{ 2 } \right] Dimensions of AB=\left[ { ML }^{ 11/2 }{ T }^{ -2 } \right]

Find the dimensional formula for {\mu}_{o} and { \varepsilon }_{ o } :

Since, k=\dfrac{1}{4\pi\epsilon_0} , dimension of [\epsilon_0]=[k^{-1}]=[ML^3T^{-4}A^{-2}]^{-1}=[M^{-1}L^{-3}T^4A^2] Also we know that , c=\dfrac{1}{\sqrt(\mu_0\epsilon_0)} this gives , dimension of [\mu_0]=[c^2\epsilon_0]^{-1}=[MLT^{-2}A^{2}]

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Important questions on Dimensions And Dimensional Analysis

Q: Out of the following the only pair that does not have identical dimensions is

Moment of inertia and moment of force doesn't have same dimension. Moment of inertia ,I= Mr^2 . And moment of force = r x F. As impulse is change in momentum so both have same dimensions.Work = F.S and torque = Fxr so both have same dimensions .Angular momentum and Planck's constant have same dimensions Angular momentum= Planck's constant/(2π) Hence, option B is correct.

Q: Find the dimensions of work.

The dimension of work is given as {\rm{work = force \times distance}} {\rm{W = Mass \times acceleration \times distance}} W = M \times L{T^{ - 2}} \times L W = M{L^2}{T^{ - 2}} The above equation shows the dimension of work done.

Q: Which of the following does not have the same dimension?

B) Pressure: \dfrac { force }{ Area } Stress : \dfrac { force }{ Area } Youngs modulus : \dfrac { stress }{ strain } , here strain is dimension less So, all three has same dimension.C) Electromotive force: \dfrac { Energy }{ charge } Potential difference: \dfrac { W}{ Q } , with is same as Electromotive Electric voltage: If you see in a capacitor, then Voltage is given by \dfrac { Energy }{ charge } D) Heat: ML^2T^{-2} Potential energy: mgh \Rightarrow ML^2T^{-2} Work done =f.d\Rightarrow ML^2T^{-2} A) Electric flux: M^{-2}SA Electric field: M^2S^{-3}A^{-1} Hence, it has quantities with different units.

Q: Find the dimensions of refractive index.

The refractive index is given as n = \dfrac{c}{v} c and v Both are speed. Hence refractive index is a dimensionless quantity having dimension {{\rm{M}}^{\rm{0}}}{{\rm{L}}^{\rm{0}}}{{\rm{T}}^{\rm{0}}}

Q: Multiple Correct Answers TypeWhich of the following pairs have the same dimensions?(L = inductance, C = capacitance, R = Resistance)

\left[ L \right] =\left[ \dfrac { \phi }{ I } \right] Where \phi is the magnetic flux and I is the current. \left[ \phi \right] =\left[ B \right] \left[ { L }^{ 2 } \right] =\left[ { MT }^{ 2 }{ A }^{ -1 } \right] { L }^{ 2 } \Rightarrow \left[ \phi \right] =\left[ { ML }^{ 2 }{ T }^{ -2 }{ A }^{ -1 } \right] So, \left[ L \right] =\left[ { MT }^{ -2 }{ L }^{ 2 }{ A }^{ -2 } \right] Now for R \left[ R \right] =\left[ \dfrac { V }{ I } \right] =\left[ \dfrac { { ML }^{ 2 }{ T }^{ -3 }{ A }^{ -1 } }{ { A }^{ -1 } } \right] \left[ R \right] =\left[ { ML }^{ 2 }{ T }^{ -3 }{ A }^{ -2 } \right] Now, \left[ \dfrac { L }{ R } \right] =\left[ \dfrac { { MT }^{ 2 }{ L }^{ 2 }{ A }^{ -2 } }{ { ML }^{ 2 }{ T }^{ -3 }{ A }^{ -2 } } \right] \left[ \dfrac { L }{ R } \right] =\left[ T \right] \left[ C \right] =\left[ \dfrac { AT }{ V } \right] =\left[ \dfrac { AT }{ { ML }^{ 2 }{ T }^{ -3 }{ A }^{ -1 } } \right] =\left[ { M }^{ -1 }{ L }^{ -2 }{ T }^{ 4 }{ A }^{ 2 } \right] So \left[ CR \right] =\left[ T \right] \left[ LC \right] =\left[ { T }^{ 2 } \right] { \left[ LC \right] }^{ 1/2 }=\left[ T \right] \left[ RC \right] =T

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

The SI unit of h is J.s or N.m.s and SI unit of c is m/s and Si unit of G is N.m^2/kg^2 For A: \sqrt{\dfrac{(N.m^2/kg^2)(m/s)}{(N.m.s)^{3/2}}}=\dfrac{kg.m^3}{N.s^5}=\dfrac{kg.m^3}{(kg.m/s^2)(s^5)}=m^2/s^3 For B: \dfrac{\sqrt{(N.m.s)(N.m^2/kg^2)}}{(m/s)^{3/2}}=\dfrac{Ns^2}{kg}=\dfrac{(kg.m/s^2)(s^2)}{kg}=m , it is the unit length. For C: \dfrac{\sqrt{(N.m.s)(N.m^2/kg^2)}}{(m/s)^{5/2}}=\dfrac{Ns^3}{m.kg}=\dfrac{(kg.m/s^2)(s^3)}{m.kg}=s For D: \sqrt{\dfrac{(N.m.s)(m/s)}{N.m^2/kg^2}}=kg

Q: Which of the following pairs of physical quantities does not have same dimensional formula?

a) Work = F \times \Delta x=[MLT^{-2}[L]=[ML^2T^{-2}]] Torque=force \times distance = [ML^2T^{-2}] b) Angular momentum mvr=[M][LT^{1}][L]=[ML^2T^{-1}] Plank's constant= \dfrac{E}{V}=\dfrac{[ML^2T^{-2}]}{[T^{-1}]}=[ML^2T^{-1}] c) Tension (force)= [MLT^{-2}] Surface tension = \dfrac{force}{length}=\dfrac{[MLT^{-2}]}{[L]}=[ML^0T^{-2}] d) Impulse =F \times \Delta t=[MLT^{-2}][T]=[MLT^{-1}] Momentum = mass \times velocity=[M][LT^{-1}]=[MLT^{-1}] So, among the above pairs only tension and surface tension does not have the same dimensional formula. They both sound similar but they both have different meaning and different applications.

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HARD

The potential energy of a particle varies the distance $x$ from a fixed origin as $U = \frac{A x}{x ^{2} + B}$ , where A and B are dimensional constants, then find the dimensional formula for AB.

View Answer

HARD

Out of the following the only pair that does not have identical dimensions is

angular momentum and Planck's constant

moment of inertia and moment of a force

work and torque

impulse and momentum

View Answer

HARD

Find the dimensions of work.

View Answer

HARD

Find the dimensional formula for $μ_{o}$ and $ε_{o}$ :

View Answer

HARD

Which of the following does not have the same dimension?

Electric flux, Electric field, Electric dipole moment

Pressure, stress, Youngs modulus

Electromotive force, Potential difference, Electric voltage

Heat, Potential energy, Work done

View Answer

HARD

Find the dimensions of refractive index.

View Answer

HARD

Two forces $P$ and $Q$ act at a point and have resultant $R$ . If $Q$ is replaced by $\frac{( R ^{2} - P ^{2} )}{Q}$ acting in the direction opposite to that of $Q$ , the resultant:

remains same

becomes half

becomes twice

none of these

View Answer

HARD

Multiple Correct Answers Type
Which of the following pairs have the same dimensions?
(L = inductance, C = capacitance, R = Resistance)

This question has multiple correct options

$\frac{L}{R}$ and CR

LR and CR

$\frac{L}{R}$ and $L C$

RC and $\frac{1}{L C}$

View Answer

HARD

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

$\frac{G c}{h ^{3 / 2}}$

$\frac{h G}{c ^{3 / 2}}$

$\frac{h G}{c ^{5 / 2}}$

$\frac{h c}{G}$

View Answer

HARD

Which of the following pairs of physical quantities does not have same dimensional formula?

Important questions on Dimensions And Dimensional Analysis

The potential energy of a particle varies the distance $x$ from a fixed origin as $U = \frac{A x}{x ^{2} + B}$ , where A and B are dimensional constants, then find the dimensional formula for AB.

Out of the following the only pair that does not have identical dimensions is

angular momentum and Planck's constant

moment of inertia and moment of a force

work and torque

impulse and momentum

Find the dimensions of work.

Find the dimensional formula for $μ_{o}$ and $ε_{o}$ :

Which of the following does not have the same dimension?

Electric flux, Electric field, Electric dipole moment

Pressure, stress, Youngs modulus

Electromotive force, Potential difference, Electric voltage

Heat, Potential energy, Work done

Find the dimensions of refractive index.

Two forces $P$ and $Q$ act at a point and have resultant $R$ . If $Q$ is replaced by $\frac{( R ^{2} - P ^{2} )}{Q}$ acting in the direction opposite to that of $Q$ , the resultant:

remains same

becomes half

becomes twice

none of these

Multiple Correct Answers Type
Which of the following pairs have the same dimensions?
(L = inductance, C = capacitance, R = Resistance)

$\frac{L}{R}$ and CR

LR and CR

$\frac{L}{R}$ and $L C$

RC and $\frac{1}{L C}$

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

$\frac{G c}{h ^{3 / 2}}$

$\frac{h G}{c ^{3 / 2}}$

$\frac{h G}{c ^{5 / 2}}$

$\frac{h c}{G}$

Which of the following pairs of physical quantities does not have same dimensional formula?

Work and torque

Angular momentum and Plancks constant

Tension and surface tension

Impulse and linear momentum

More important questions

Important questions on Dimensions And Dimensional Analysis

The potential energy of a particle varies the distance x from a fixed origin as U=x2+BAx​​, where A and B are dimensional constants, then find the dimensional formula for AB.

Out of the following the only pair that does not have identical dimensions is

angular momentum and Planck's constant

moment of inertia and moment of a force

work and torque

impulse and momentum

Find the dimensions of work.

Find the dimensional formula for μo​ and εo​:

Which of the following does not have the same dimension?

Electric flux, Electric field, Electric dipole moment

Pressure, stress, Youngs modulus

Electromotive force, Potential difference, Electric voltage

Heat, Potential energy, Work done

Find the dimensions of refractive index.

Two forces P and Q act at a point and have resultant R. If Q is replaced by Q(R2−P2)​ acting in the direction opposite to that of Q, the resultant:

remains same

becomes half

becomes twice

none of these

Multiple Correct Answers Type Which of the following pairs have the same dimensions? (L = inductance, C = capacitance, R = Resistance)

RL​ and CR

LR and CR

RL​ and LC​

RC and LC1​

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

h3/2Gc​​

c3/2hG​​

c5/2hG​​

Ghc​​

Which of the following pairs of physical quantities does not have same dimensional formula?

Work and torque

Angular momentum and Plancks constant

Tension and surface tension

Impulse and linear momentum

More important questions

The potential energy of a particle varies the distance $x$ from a fixed origin as $U = \frac{A x}{x ^{2} + B}$ , where A and B are dimensional constants, then find the dimensional formula for AB.

Find the dimensional formula for $μ_{o}$ and $ε_{o}$ :

Two forces $P$ and $Q$ act at a point and have resultant $R$ . If $Q$ is replaced by $\frac{( R ^{2} - P ^{2} )}{Q}$ acting in the direction opposite to that of $Q$ , the resultant:

Multiple Correct Answers Type
Which of the following pairs have the same dimensions?
(L = inductance, C = capacitance, R = Resistance)

$\frac{L}{R}$ and CR

$\frac{L}{R}$ and $L C$

RC and $\frac{1}{L C}$

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

$\frac{G c}{h ^{3 / 2}}$

$\frac{h G}{c ^{3 / 2}}$

$\frac{h G}{c ^{5 / 2}}$

$\frac{h c}{G}$