H C Verma Solutions for Chapter: Introduction to Physics, Exercise 4: EXERCISES

Find the dimensions of

(a) electric dipole moment, $p$ and

(b) magnetic dipole moment, $M$.

The defining equations are $p=q·d$ and $M=I A$, where $d$ is distance, $A$ is area, $q$ is charge and $I$ is current.

Find the dimensions of
(a) the specific heat capacity $c$,

(b) the coefficient of linear expansion $\alpha$ and

(c) the gas constant $R$.

Some equations involving these quantities are $Q=mc\left(T_2-T_1\right), l_t=l_0\left[1+\alpha \left(T_2-T_1\right)\right]$ and $PV=nRT$.

Taking force, length and time to be the fundamental quantities find the dimensions of
(a) density,
(b) pressure,
(c) momentum and
(d) energy.

The average speed of a snail is $0.020$ $\mathrm{miles} {\mathrm{hour}}^{-1}$ and that of a leopard is $70$ $\mathrm{miles} {\mathrm{hour}}^{-1}$. Convert these speeds in $\mathrm{SI}$ units. $\left(1 \mathrm{mile}=1.6 \mathrm{km}\right)$
 

The height of mercury column in a barometer in a Calcutta laboratory was recorded to be $75 \mathrm{cm}$. Calculate this pressure in $SI$ and $CGS$ units using the following data: Specific gravity of mercury$=13.6,$ density of water $={10}^{3 }\mathrm{kg} {\mathrm{m}}^{-3}, g=9.8 \mathrm{m} {\mathrm{s}}^{-2}$ at Calcutta. Pressure $=h\rho g$ in usual symbols.
 

The theory of relativity reveals that mass can be converted into energy. The energy $E$ so obtained is proportional to certain powers of mass $m$ and $c$ is the speed of light. Guess a relation among the quantities using the method of dimensions.

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 the dimensional analysis?

Test if the equations are dimensionally correct,

$\left(a\right) h=\frac{2S\mathrm{cos\theta }}{\rho rg}$

$\left(b\right) v=\sqrt{\frac{P}{\rho }}$

$\left(c\right) V=\frac{\mathrm{\pi }P{r}^{4}t}{8\eta l}$

$\left(d\right) v=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{mgl}{I}}$

where $h=$height, $S=$surface tension, $\rho =$density, $P=$pressure, $V=$volume, $\eta =$coefficient of viscosity, $v=$frequency and $I=$moment of inertia.