Question

If v = wA cos ( wt -kx) , then find the dimension of k/w ​

Answer 1

Need to FinD :-

• The dimension of k/ω .

$\red{\frak{Given}}\Bigg\{ \sf v = \omega A \ cos ( \omega t - kx )$

$\red{\bigstar}\textsf{\textbf{We know that Dimension is :- }}$

• It refers to the power given to the fundamental quantities in the dimensional formula of a given quantity .

Step 1: Here ωt must be angle :-

We know that angle is dimensionless . So that ,

$\sf\dashrightarrow \omega t = [ M^0L^0T^0] \\\\\\\sf\dashrightarrow \omega =\dfrac{[ M^0L^0T^0]}{[T]}\\\\\\\sf\dashrightarrow \omega = [ M^0L^0T^{0-1}] \\\\\\\sf\dashrightarrow\red{ \omega =[ T^{-1}] }$

$\rule{200}2$

Step 2: Similarly kx is also a angle :-

$\sf\dashrightarrow kx =[ M^0L^oT^0] \\\\\\\sf\dashrightarrow k = \dfrac{[ M^0L^0T^0]}{[L]} \\\\\\\sf\dashrightarrow\red{ k = [ L^{-1}]}$

$\rule{200}2$

Step 3 : Finding the Ratio of k/ω :-

$\sf\dashrightarrow \dfrac{k}{x}=\dfrac{L^{-1}}{T^{-1}} \\\\\\\sf\dashrightarrow \boxed{\pink{\sf \dfrac{k}{x}= [ L^{-1} T ] }}$

Hence the required answer is [ L-¹ T ]