Question

The equation of a wave, travelling with amplitude A, linear u, angular velocity is given by $ y= A\sin \omega (\dfrac{x}{v}-k) $
The dimensions of 'k' are
(A) $ [ $ T $ ] $
(B) $ [ $ $ T^{-1} $ $ ] $
(C) $ [ $ $ T^{-2} $ $ ] $
(D) $ [ $ $ T^{2} $ $ ] $

Answer 1

Hint
Here we will use 2 properties of dimensional homogeneity properties where the dimension of the right side is always equal to the left side in any equation and we can only substrate or add two or more quantities if they have the same dimension. An equation could be dimensionally homogeneous but invalid if the equation is also not fully-balanced example 4T=T.

Complete step by step answer
Dimension for linear velocity v= $ LT^{-1} $
Dimension for displacement x = L
According to dimensional homogeneity,
 $\Rightarrow \dfrac{x}{v} $ have same dimension of k as they have substrate to each other
Dimensional formula for k is:
 $\Rightarrow \lbrack k\rbrack =\lbrack \dfrac{x}{v}\rbrack $
Substituting the value of dimension of x and v we get;
 $\Rightarrow \lbrack \dfrac{L}{LT^{-1}}\rbrack $
After solving the numerator and denominator we get ,
 $\Rightarrow \lbrack k\rbrack =\lbrack T\rbrack $
Hence, Correct option is (A).

Note
Dimensions of a physical quantity are the powers to which the fundamental units are raised to gain one unit of that quantity. Dimensional evaluation is the process of checking the relations between physical quantities by identifying the dimensions of the physical quantities. These dimensions are independent of the numerical multiples and constants. If Q is the unit of a derived quantity represented by Q = M $ ^{a} $ L $ ^{b} $ T $ ^{c} $, then M $ ^{a} $ L $ ^{b} $ T $ ^{c} $ is called dimensional formula and the exponents a, b and, c is called the dimensions. Dimensional analysis has various applications like to change units from one system to another, consistency of a dimensional equation. There are 7 primary dimensions which are mass, length, time, temperature, electric current, amount of light, and amount of matter.