Limitation of Dimensional Analysis

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

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

Q. The time dependence of physical quantity $P$ is found to be of the form $P=P_o{e}^{-\alpha t^2}$, where $‘t^{\prime }$ is the time and $\alpha$ is some constant. Then the constant $\alpha$ will

1. be dimensionless
2. have dimensions of $T^{-2}$
3. have dimensions of $P$
4. have dimensions of $P$ multiplied by $T^{-2}$

Q. In a simple harmonic oscillation, what fraction of total mechanical energy is in the form of kinetic energy, when the particle is midway between mean and extreme position.

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

Q. The dimension of $a$ in Van der Waals equation $(P+\frac{a{n}^2}{V^2})(V-nb)=nRT$ is

1. $\left[M{L}^5{T}^{-2}\right]$
2. $\left[M{L}^{5}T\right]$
3. $\left[M^{-1}{L}^5{T}^2\right]$
4. $\left[M{L}^{-5}{T}^{-1}\right]$

Q. The Van Der Waal’s equation of state for some gases can be expressed as: $(P+\frac{a}{V^2})(V-b)=RT$ Where $P$ is the pressure, $V$ is the molar volume and $T$ is the absolute temperature of the given sample of gas and $a,b$ and $R$ are constants.

Which of the following does not have the same dimension formula as that of $RT$ ?

1. $PV$
2. $Pb$
3. $\frac{a}{V^2}$
4. $\frac{ab}{V^2}$

Q.

Define dimensional formula. Give uses of dimensional analysis. Write down the limitations.

Q.

Capillary tube is a very thin tube. If it is dipped in water, water rises upto height $h=\frac{2T}{r\rho g},$ where r is the radius of tube, $\rho$ is density, and g is acceleration due to gravity. Find dimensional formula of [T]

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

Q. The potential energy of a particle varies with distance $x$ as $\frac{A{x}^{1/2}}{x^2+B}$, where $A$ and $B$ are constants. The dimensional formula for $A\times B$ is

1. $M^1{L}^{\frac{7}{2}}{T}^{-2}$
2. $M^1{L}^{\frac{11}{2}}{T}^{-2}$
3. $M^1{L}^{\frac{9}{2}}{T}^{-2}$
4. $M^1{L}^{\frac{5}{2}}{T}^{-2}$

Q.

Mention any two limitations of dimensional analysis.

Q. The Van Der Waal’s equation of state for some gases can be expressed as: $(P+\frac{a}{V^2})(V-b)=RT$ Where $P$ is the pressure, $V$ is the molar volume and $T$ is the absolute temperature of the given sample of gas and $a,b$ and $R$ are constants.

The dimensions of constant $b$ are

1. $M{L}^5{T}^{-2}$
2. $M{L}^{-1}{T}^{-2}$
3. $L^3$
4. $L^6$

Q. Write important limitations of dimensional analysis.

Q. State any two limitations of dimensional analysis.

Q. A quantity $x$ is given by $x=\frac{IF{v}^2}{W{L}^4}$, in terms of moment of inertia $I$, force $F$, velocity $v$, work $W$ and length $L$. The dimensional formula for $x$ is the same as that of :

1. planck’s constant
2. force constant
3. energy density
4. coefficient of viscosity

Q. The relation $P=\frac{\alpha }{\beta }{e}^{\left(\frac{-\alpha z}{k\theta }\right)}$, where $p$ is pressure, $z$ is distance, $k$ is Bolzmann constant and $\theta$ is temperature. The dimensional formula of $\beta$ will be

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

Q. Answer the following:
Mention three limitations of dimensional analysis.

Q.

Test dimensionally if the equation $v^2=u^2$ + 2ax may be correct.

1. correct for sure

2. may be correct

3. wrong for sure

4. none of these

Q.

The force F is given in terms of time t and displacement x by the equation F = A cos Bx + C sin Dt. The dimensional formula of $\frac{D}{B}$ is

1. $\left[M^0{L}^0{T}^0\right]$

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

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

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

Q. The Van Der Waal’s equation of state for some gases can be expressed as: $(P+\frac{a}{V^2})(V-b)=RT$
Where $P$ is the pressure, $V$ is the molar volume and $T$ is the absolute temperature of the given sample of gas and $a,b$ and $R$ are constants.

The dimensions of $a$ are

1. $L^3$
2. $M{L}^5{T}^{-2}$
3. $M{L}^{-1}{T}^{-2}$
4. $L^6$

Q. A highly rigid cubical block $A$ of small mass $M$ and side $L$ is fixed rigidly onto another cubical block $B$ of the same dimensions and of modulus of rigidity $\eta$ such that the lower face of $A$ completely covers the upper face of $B$. The lower face of $B$ is rigidly held on a horizontal surface. A small force $F$ is applied perpendicular to one of the side faces of $A$. After the force is withdrawn block $A$ executes small oscillations. The time period of which is given by

1. $2\pi \sqrt{\frac{L}{M\eta }}$
2. $2\pi \sqrt{\frac{M\eta }{L}}$
3. $2\pi \sqrt{\frac{ML}{\eta }}$
4. $2\pi \sqrt{\frac{M}{\eta L}}$

Q. Two very long co-axial cylinders of radius R and 2R carry current I in opposite directions. Then energy stored in cylindrical region of radius 4R of unit length is

1. $\frac{3}{8}\frac{{\mu }_0{i}^2}{\pi }$
   
2. $\frac{{\mu }_0{i}^2}{4\pi }ln\left(2\right)$
   
3. $\frac{{\mu }_0{i}^2}{4\pi }ln\left(4\right)$
   
4. zero

Q.

Q. The time dependence of physical quantity $P$ is found to be of the form $P=P_o{e}^{-\alpha t^2}$, where $‘t^{\prime }$ is the time and $\alpha$ is some constant. Then the constant $\alpha$ will

1. be dimensionless
2. have dimensions of $T^{-2}$
3. have dimensions of $P$
4. have dimensions of $P$ multiplied by $T^{-2}$

Q.

As the sir said that he will teach the extent version of dimensions analysis but there is no vedio about it

Please share the the information to me

Q. Assertion :The units of some physical quantities can be expressed as combination of the base units. Reason: We need only a limited number of units for expressing the derived physical quantities.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. If the linear charge density of a cylinder is $4\mu C{m}^{-1}$ then electric field intensity at point $3.6cm$ from axis is.

1. $2\times {10}^{6}N{C}^{-1}$
2. $12\times {10}^{7}N{C}^{-1}$
3. $4\times {10}^{5}N{C}^{-1}$
4. $8\times {10}^{7}N{C}^{-1}$

Q. Taking the electronic charge as ‘e’ and the permittivity as $‘\epsilon 0^{\prime }$, use dimensional analysis to determine the correct expression for $\omega p$

1. $\sqrt{\frac{Ne}{m{\epsilon }_0}}$
2. $\sqrt{\frac{m{\epsilon }_0}{N{e}^2}}$
3. $\sqrt{\frac{N{e}^2}{m{\epsilon }_0}}$
4. $\sqrt{\frac{m{\epsilon }_0}{Ne}}$

Q. The time dependence of a physical quantity $P$ is given by $P=P_0{e}_{\alpha }(-\alpha t^2)$, where $\alpha$ is a constant and $t$ is time. The constant $\alpha$

1. Is a dimension less
2. Has dimensions of $P$
3. Has dimensions of $T$
4. Has dimensions of $T^{-2}$

Q. The relation $P=\frac{\alpha }{\beta }{e}^{\left(\frac{-\alpha z}{k\theta }\right)}$, where $p$ is pressure, $z$ is distance, $k$ is Bolzmann constant and $\theta$ is temperature. The dimensional formula of $\beta$ will be

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

Q. Which of the following equations cannot be deduced using dimensional analysis?

Q. Sir what is meant by dimensional analysis and its limitations .
Can I have a answer in easy language. Not understood any of the dimensional analysis ?