Dimensional Analysis

Q.

The frequency of vibration of a string depends on the length L between the nodes, the tension F in the string and its mass per unit length m, Guess the expression for its frequency from dimensional analysis.

Q.

What is the dimensional formula of epsilon not?

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

1. $\sqrt{\frac{hG}{C^3}}$
2. $\sqrt{\frac{hG}{C^{5/2}}}$
3. $\sqrt{\frac{hC}{G}}$
4. $\sqrt{\frac{GC}{h^{3/2}}}$

Q. A charge Q is fixed at each of two opposite corners of a square. A charge q is placed at each of the other two corners. If the resultant electric force on Q is zero, then Q and q are related as

1. $Q=\sqrt{2}q$
2. $Q=-2\sqrt{2}q$
3. $Q=Q=-\sqrt{2}q$
4. $Q=2\sqrt{2}q$

Q. If dimensions of critical velocity $v_c$ of a liquid flowing through a tube are expressed as $\left[{\eta }^x{\rho }^y{r}^z\right]$, where $\eta ,\rho$ and $r$ are the coefficient of viscosity of liquid, density of liquid and radius of the tube respectively, then values of $x,y$ and $z$ are given by

1. 1, -1, -1
2. 1, 1, 1
3. 
   -1, -1, 1
4. -1, -1, -1

Q.

Assuming that the mass M of the largest stone that can be moved by flowing river depends on the velocity of water V density of water p and the acceleration due to gravity G then M is directly proportional to

Q. The equation of a wave is given by
$y=A\sin \omega (\frac{x}{v}-a)$, where $\omega$ is the angular velocity and $v$ is the linear velocity of the wave. The dimensions of $a$ is

1. $\left[M\right]$
2. $\left[T\right]$
3. $\left[L\right]$
4. $\left[K\right]$

Q.

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

1. $F{A}^2{V}^{-3}$

2. $F{A}^2{V}^{-2}$

3. $F{A}^{-1}{V}^0$

4. $F{A}^2{V}^{-1}$

Q. Consider the region of electric field, as shown in the figure. Work done by the electric field, in carrying a protonfrom point B to point A, will be4.VA 10 VVB 20 V(1) 1.610-18J(3) 1.6 x 10-19 J(2)(4)-1.610-18J-1.6 x 10-19 Jsh Educational Saricoo Lt

Q. The given figure shows a set of equipotential surfaces. Find the magnitude and direction of electric field that exists in the region.

Q.

A cube of side $L$ is placed in a uniform electric field, $\overrightarrow{E}=E\hat{i}$. The net electric flux through the cube is

1. 0
2. $L^{2}E$
3. $4L^{2}E$
4. $6L^{2}E$

Q.

Find the dimension formula for resistance.

Q. If time $\left(t\right)$, velocity $\left(v\right)$, and angular momentum $\left(l\right)$ are taken as the fundamental units. Then the dimension of mass $\left(m\right)$ in terms of $t,v$ and $l$ is :

1. $\left[t^{-2}{v}^{-1}{l}^1\right]$
2. $\left[t^{-1}{v}^1{l}^{-2}\right]$
3. $\left[t^{-1}{v}^{-2}{l}^1\right]$
4. $\left[t^1{v}^2{l}^{-1}\right]$

Q. What is the SI unit of surface tension?

Q. What is the dimensional formula for permittivity of free space?

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

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.

What is the SI unit of mechanical equivalent of heat?

Q. Define power.

Q. In a series resonant $LCR$ circuit, the voltage across resistor $R$ is $100\text{V}$ and $R=1\text{k}\Omega$ with $C=2\mu \text{F}$. The resonant frequency $\omega$ is $200\text{rad/s}$. At resonance, the voltage across $L$ is

1. $2.5\times {10}^{-2}\text{V}$
2. $40\text{V}$
3. $250\text{V}$
4. $4\times {10}^{-3}\text{V}$

Q. In C.G.S. system the magnitude of the force is 100 dynes. In another system where the fundamental physical quantities are kilogram, metre and minute, the magnitude of the force is

1. 3.6
2. 36
3. 0.036
4. 0.36

Q. A particle executes simple harmonic motion with an amplitude of $5\text{cm}$. When the particle is at $4\text{cm}$ from the mean position, the magnitude of its velocity in SI units is equal to that of its acceleration. Then, its periodic time in seconds is

1. $\frac{4\pi }{3}$
2. $\frac{3}{8}\pi$
3. $\frac{8\pi }{3}$
4. $\frac{7}{3}\pi$

Q. If force $\left(F\right)$, velocity $\left(V\right)$ and time $\left(T\right)$ are taken as fundamental units, then dimensions of mass are

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

Q.

What is the dimensional formula for thermal resistance?

Q.

A student performs experiment with simple pendulum and measures time for 10 vibrations . If he measures time for 100 vibrations , the relative error in measurements of time period will be reduced by

Q. The unit of plane angle is radian, hence its dimensions are

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

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. The electric field in a region is given by $\overrightarrow{E}=(Ax+B)\hat{i}$, where $E$ is in ${\text{NC}}^{-1}$ and $x$ is in metres. The values of constants are $A=20$ SI unit and $B=10$ SI unit. If the potential at $x=1$, is $V_1$, and that at $x=-5$ is $V_2$, then $V_1-V_2$

1. $-48\text{V}$
2. $180\text{V}$
3. $320\text{V}$
4. $-520\text{V}$

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

Q.

A long solenoid with 15 turns per cm has a small loop of area 2.0 cm² placed inside the solenoid normal to its axis. If the current carried by the solenoid changes steadily from 2.0 A to 4.0 A in 0.1 s, what is the induced emf in the loop while the current is changing?

Q.

Which of the following is not a fundamental quantity?

1. Current

2. Time

3. Weight

4. Length