Embibe Experts Solutions for Chapter: Physical World and Measurement, Exercise 4: Exercise-4

The position of a particle at time $t$ is given by the relation $x\left(t\right)=\left(\frac{v_0}{\alpha }\right)\left(1-e^{-\alpha t}\right)$, $v_0$ where is a constant and $\alpha >0$. Find the dimensions of $v_0$ and $\alpha$.

The related equation are: $\mathrm{Q}=\mathrm{mc}\left({\mathrm{T}}_2-{\mathrm{T}}_1\right),{\mathrm{\ell }}_1={\mathrm{\ell }}_0\left[1+\mathrm{\alpha }\left({\mathrm{T}}_2-{\mathrm{T}}_1\right)\right]$ and $\mathrm{PV}=\mathrm{nRT}$, where the symbols have their usual meanings. Find the dimensions of

(A) specific heat capacity $\left(\mathrm{C}\right)$

(B) coefficient of linear expansion $\left(\mathrm{\alpha }\right)$ and

(C) the gas constant $\left(\mathrm{R}\right)$

A particle of mass $\mathrm{m}$ is in a uni-directional potential field where the potential energy of a particle depends on the $\mathrm{x}$- coordinate given by ${\mathrm{\varphi }}_{\mathrm{x}}={\mathrm{\varphi }}_0\left(1-\mathrm{cosax}\right)$ and '${\mathrm{\varphi }}_0$' and '$\mathrm{a}$' constants. Find the physical dimensions of '$\mathrm{a}$' and ${\mathrm{\varphi }}_0$.

In the formula, $p=\frac{nRT}{V-b}{e}^{\frac{a}{RTV}},$ find the dimensions of $a$ and $b,$ where $p=$pressure, $n=$number of moles, $T=$temperature, $V=$volume and $R=$universal gas constant.

If instead of mass, length and time as fundamental quantities we choose velocity, acceleration and force as fundamental quantities and express their dimensional symbols as $\mathrm{v}, \mathrm{a}$ and $\mathrm{F}$ respectively. Show that the dimensions of Young's modulus can be expressed as $\left[{\mathrm{Fa}}^2{\mathrm{v}}^{-4}\right]$.

If energy $\left(\mathrm{E}\right)$, velocity $\left(\mathrm{v}\right)$ and time $\left(\mathrm{T}\right)$ are fundamental units then the dimension of surface tension is $\left[{\mathrm{Ev}}^{-\mathrm{\alpha }}{\mathrm{T}}^{-\mathrm{\beta }}\right]$. Find $\alpha -\beta$

In a system called the star system we have $1$ star kilogram $={10}^{20} \mathrm{kg}$. $1$ star meter $={10}^8 \mathrm{m}$, $1$star second $={10}^3$ second. If the value of $1$ joule in this system is ${10}^{-10\alpha } \mathrm{star} \mathrm{joule}$, then find $\alpha$ .

A new unit of length is chosen such that the speed of light in vacuum is unity. What is the distance between the Sun and Earth in terms of the new unit if light takes $8 \min$ and $20 \mathrm{s}$ to cover this distance?