Unit And Dimensions Set-2

Question 1

If $E$ and $H$ represents the intensity of electric field and magnetising field respectively, then the unit of $E/H$ will be:

Accepted Answer

Ohm

Answer

Joule

Answer

Newton

Answer

Mho

Question 2

A physical quantity $\vec{S}$ is defined as $\vec{S} = (\vec{E} 	imes \vec{B}) / \mu_0$, where $\vec{E}$ is electric field, $\vec{B}$ is magnetic field and $\mu_0$ is the permeability of free space. The dimensions of $\vec{S}$ are the same as the dimensions of which of the following quantity (ies):

Accepted Answer

Force

Answer

Energy

Answer

Charge $	imes$ Current

Answer

Length $	imes$ Time

Question 3

The density of a material in SI units is 128 kg m$^{-3}$. In certain units in which the unit of length is 25 cm and the unit of mass 50 g, the numerical value of density of the material is

Accepted Answer

40

Answer

16

Answer

640

Answer

410

Question 4

The dimensional formula of latent heat is:

Accepted Answer

$[M^0 L^2 T^{-2}]$

Answer

$[M L^2 T^{-2}]$

Answer

$[M L T^{-2}]$

Answer

$[M L^2 T^{-1}]$

Question 5

If $\epsilon_0$ is the permittivity of free space and $E$ is the electric field, then $\epsilon_0 E^2$ has the dimensions:

Accepted Answer

$[M L^{-1} T^{-2}]$

Answer

$[M L^{-2} T^{-2} A^2]$

Answer

$[M^{-1} L^{-3} T^4 A^2]$

Answer

$[M L^2 T^{-2}]$

Question 6

Given below are two statements:
Statement (I): Dimensions of specific heat is $[L T^{-2} K^{-1}]$
Statement (II): Dimensions of gas constant is $[M L T^{-1} K^{-1}]$

Accepted Answer

Statement (I) is correct but statement (II) is incorrect

Answer

Statement (I) is incorrect but statement (II) is correct

Answer

Both statement (I) and statement (II) are incorrect

Answer

Both statement (I) and statement (II) are correct

Question 7

What is the dimensional formula of $a b^{-1}$ in the equation $\left( p + \frac{a}{V^2} \right) (V - b) = RT$, where letters have their usual meaning.

Accepted Answer

$[M L T^{-2}]$

Answer

$[M L^5 T^{-2}]$

Answer

$[M L^5 T^{-3}]$

Answer

$[M L^5 T^{-4}]$

Question 8

If $G$ be the gravitational constant and $u$ be the energy density then which of the following quantity has the dimension as that of $\sqrt{u G}$:

Accepted Answer

Force per unit mass

Answer

Pressure gradient per unit mass

Answer

Gravitational potential

Answer

Energy per unit mass

Question 9

The dimensional formula of angular impulse is:

Accepted Answer

$[M L^2 T^{-1}]$

Answer

$[M L^2 T^{-2}]$

Answer

$[M L T^{-1}]$

Answer

$[M L^{-2} T^{-1}]$

Question 10

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.

Accepted Answer

Statement-I is true but Statement-II is false

Answer

Both Statement-I and Statement-II are false

Answer

Both Statement-I and Statement-II are true

Answer

Statement-I is false but Statement-II is true

Question 11

Applying the principle of homogeneity of dimension, which one is correct? where $T$ is time period, $G$ is gravitational constant, $M$ is mass, $r$ is radius of orbit.

Accepted Answer

$T^2 = \frac{4\pi^2 r^3}{G M}$

Answer

$T^2 = \frac{4\pi^2 r}{G M^2}$

Answer

$T^2 = \frac{4\pi^2 r^2}{G M}$

Answer

$T^2 = \frac{4\pi^2 r^2}{G M^2}$

Question 12

A force is represented by $F = a x^2 + b t^{1/2}$ where $x$ = distance and $t$ = time. The dimensions of $b^2 / a$ are:

Accepted Answer

$[M L^3 T^{-3}]$

Answer

$[M L^2 T^{-2}]$

Answer

$[M L^2 T^{-1}]$

Answer

$[M L^2 T^{-3}]$

Question 13

If mass is written as $m = k c^P G^{-1/2} h^{1/2}$ then the value $P$ will be: (Constants have their usual meaning with $k$ dimensionless constant)

Accepted Answer

1/2

Answer

1/3

Answer

2

Answer

-1/3

Question 14

The equation of state of a real gas is given by $\left(P + \frac{a}{V^2}\right)(V - b) = RT$ where $P$, $V$ and $T$ are pressure volume and temperature respectively and $R$ is universal gas constant. The dimensions of $\frac{a}{b^2}$ is similar to that of:

Accepted Answer

$P$

Answer

$PV$

Answer

$RT$

Answer

$R$

Question 15

The frequency $
u$ of an oscillating liquid drop may depend upon radius $r$ of the drop, density $ho$ of liquid and the surface tension $s$ of the liquid as: $
u = r^a ho^b s^c$. The values of $a, b$ and $c$ respectively are

Accepted Answer

$\left( -\frac{3}{2}, -\frac{1}{2}, \frac{1}{2} \right)$

Answer

$\left( -\frac{3}{2}, \frac{1}{2}, \frac{1}{2} \right)$

Answer

$\left( \frac{3}{2}, -\frac{1}{2}, \frac{1}{2} \right)$

Answer

$\left( -\frac{3}{2}, -\frac{1}{2}, -\frac{1}{2} \right)$

Question 16

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

Accepted Answer

$[t^{-1} v^{-2} l^1]$

Answer

$[t^{-2} v^{-1} l^1]$

Answer

$[t^{-1} v^1 l^{-2}]$

Answer

$[t^1 v^2 l^{-1}]$

Question 17

In a typical combustion engine the work done by a gas molecule is given by $W = \alpha^2 \beta e^{-\frac{\alpha x^2}{k T}}$, where $x$ is the displacement, $k$ is the Boltzmann constant and $T$ is the temperature. If $\alpha$ and $\beta$ are constants, dimensions of $\alpha$ will be:

Accepted Answer

$[M^0 L T^0]$

Answer

$[M L T^{-1}]$

Answer

$[M^2 L T^{-2}]$

Answer

$[M L T^{-2}]$

Question 18

Dimensional formula for thermal conductivity is (here $K$ denotes the temperature):

Accepted Answer

$M L T^{-3} K^{-1}$

Answer

$M L T^{-2} K^{-2}$

Answer

$M L T^{-2} K$

Answer

$M L T^{-3} K$

Question 19

Expression for time in terms of $G$ (universal gravitational constant), $h$ (Planck constant) and $c$ (speed of light) is proportional to:

Accepted Answer

$\sqrt{\frac{G h}{c^5}}$

Answer

$\sqrt{\frac{h c^5}{G}}$

Answer

$\sqrt{\frac{c^3}{G h}}$

Answer

$\sqrt{\frac{G h}{c^3}}$

Question 20

Young's modulus is determined by the equation given by $Y = 49000 \frac{m}{\ell}$ where $m$ is the mass and $\ell$ is the extension of wire used in the experiment. Now error in Young modulus (Y) is estimated by taking data from $m-\ell$ plot in graph paper. The smallest scale divisions are 5 g and 0.02 cm along load axis and extension axis respectively. If the value of $m$ and $\ell$ are 500 g and 2 cm respectively then percentage error of $Y$ is:

Accepted Answer

2%

Answer

0.2%

Answer

0.02%

Answer

0.5%

Question 21

To find the spring constant ($k$) of a spring experimentally, a student commits 2% positive error in the measurement of time and 1% negative error in measurement of mass. The percentage error in determining value of $k$ is:

Accepted Answer

5%

Answer

3%

Answer

4%

Answer

2%

Question 22

Time periods of oscillation of the same simple pendulum measured using four different measuring clocks were recorded as 4.62 s, 4.632 s, 4.6 s and 4.64 s. The arithmetic mean of these reading in correct significant figure is:

Accepted Answer

4.62 s

Answer

4.623 s

Answer

4.6 s

Answer

5 s

Question 23

In an experiment to measure focal length ($f$) of convex lens, the least counts of the measuring scales for the position of object ($u$) and for the position of image ($v$) are $\Delta u$ and $\Delta v$, respectively. The error in the measurement of the focal length of the convex lens will be:

Accepted Answer

$f^2 \left[ \frac{\Delta u}{u^2} + \frac{\Delta v}{v^2} \right]$

Answer

$\frac{\Delta u}{u} + \frac{\Delta v}{v}$

Answer

$2f \left[ \frac{\Delta u}{u} + \frac{\Delta v}{v} \right]$

Answer

$f \left[ \frac{\Delta u}{u} + \frac{\Delta v}{v} \right]$

Question 24

The radius $r$, length $l$ and resistance $R$ of a metal wire was measured in the laboratory as $r = (0.35 \pm 0.05)$ cm, $R = (100 \pm 10)$ ohm, $l = (15 \pm 0.2)$ cm. The percentage error in resistivity of the material of the wire is:

Accepted Answer

39.9%

Answer

25.6%

Answer

37.3%

Answer

35.6%

Question 25

The measured value of the length of a simple pendulum is 20 cm with 2 mm accuracy. The time for 50 oscillations was measured to be 40 seconds with 1 second resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is $N\%$. The value of $N$ is:

Accepted Answer

6

Answer

4

Answer

8

Answer

5

Question 26

If the percentage errors in measuring the length and the diameter of a wire are 0.1% each. The percentage error in measuring its resistance will be:

Accepted Answer

0.3%

Answer

0.2%

Answer

0.1%

Answer

0.144%

Question 27

The resistance $R = \frac{V}{I}$ where $V = (200 \pm 5)V$ and $I = (20 \pm 0.2)A$, the percentage error in the measurement of $R$ is:

Accepted Answer

3.5%

Answer

7%

Answer

3%

Answer

5.5%

Question 28

A physical quantity $Q$ is found to depend on quantities $a, b, c$ by the relation $Q = \frac{a^4 b^3}{c^2}$. The percentage error in $a, b$ and $c$ are 3%, 4% and 5% respectively. Then, the percentage error in $Q$ is:

Accepted Answer

34%

Answer

66%

Answer

43%

Answer

14%

Question 29

Two resistances are given as $R_1 = (10 \pm 0.5) \Omega$ and $R_2 = (15 \pm 0.5) \Omega$. The percentage error in the measurement of equivalent resistance when they are connected in parallel is

Accepted Answer

4.33

Answer

6.33

Answer

2.33

Answer

5.33

Question 30

If $Z = \frac{A^2 B^3}{C^4}$, then the relative error in $Z$ will be:

Accepted Answer

$\frac{2\Delta A}{A} + \frac{3\Delta B}{B} + \frac{4\Delta C}{C}$

Answer

$\frac{\Delta A}{A} + \frac{\Delta B}{B} + \frac{\Delta C}{C}$

Answer

$\frac{\Delta A}{A} + \frac{\Delta B}{B} - \frac{\Delta C}{C}$

Answer

$\frac{2\Delta A}{A} + \frac{3\Delta B}{B} - \frac{4\Delta C}{C}$

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