Question

Find the dimension of \[\dfrac{a}{b}\] from the expression $V = a + b\sin (\omega t + \phi )$ where, V is the velocity and ‘t’ is time.
A. ${M^0}{L^0}{T^0}$
B. ${M^1}{L^1}{T^0}$
C. ${M^2}{L^1}{T^0}$
D. ${M^1}{L^2}{T^1}$

Answer 1

Hint: In physics, every physical quantity can be expressed in terms of seven basic fundamental physical quantities which are Mass (M) Length (L) Time (T) Current (A) Amount of substance (mole) Luminous intensity (candela) Temperature (kelvin) and these notations are used to form a dimensions of any physical quantity in the form of $[{M^a}{L^b}{T^c}]$ only those quantities are expressed in dimensional formula on which the given physical quantity depends mathematical wise.

Complete step by step answer:
According to the question, we have given that an expression $V = a + b\sin (\omega t + \phi )$ Since, according to dimensional laws, two physical quantities having same dimensions can only be added or subtracted together to form a new physical quantity having same dimensions as of individual physical quantities which are added or subtracted.

Here, $V$ is the velocity which is the ratio of distance and time so, dimensions of Velocity are expressed as $[{M^0}L{T^{ - 1}}]$. Now, since $V$ is the sum of two quantities which are $a$ and $b\sin (\omega t + \phi )$ so, these individuals will also be of same dimension of velocity such that dimension of a will be $[{M^0}L{T^{ - 1}}]$.

since, $\sin (\omega t + \phi )$ is a trigonometric ratio which has no dimensions so the quantity $b\sin (\omega t + \phi )$ only depend upon dimension of b and which will be same as dimension of velocity such that dimension of $b$ will be $[{M^0}L{T^{ - 1}}]$. Now, divide the quantity $a$ and $b$ dimension formula wise we get,
$\dfrac{a}{b} = \dfrac{{[{M^0}L{T^{ - 1}}]}}{{[{M^0}L{T^{ - 1}}]}}$
$\therefore \dfrac{a}{b} = [{M^0}{L^0}{T^0}]$

Hence, the correct option is A.

Note: It should be remembered that, trigonometric functions and many mathematical constants such as $\pi ,e$ have no dimensions and known as dimensionless quantities, here the quantity $\dfrac{a}{b}$ is also a dimensionless quantity. Also, when a constant is multiplied with a physical quantity the dimension of that physical quantity remains unchanged.