Question

If energy E, velocity v and time T are taken as fundamental quantities, the dimensional formula for surface tension is

• A. $ [E{{v}^{-2}}{{T}^{-2}}] $
• B. $ [{{E}^{2}}v{{T}^{-2}}] $
• C. $ [E{{v}^{-2}}{{T}^{-1}}] $
• D. $ [{{E}^{-2}}{{v}^{2}}{{T}^{-1}}] $

Answer 1

The Correct Option is A

The dimensions of given fundamental quantities are given below: Energy $ =[M{{L}^{2}}{{T}^{-2}}] $ Velocity $ (v)=(L{{T}^{-1}}) $ Time $ (T)=[T] $ and surface tension $ [S]=[M{{T}^{-2}}] $ Let $ [S]\propto {{[E]}^{a}}\times {{[v]}^{b}}\times {{[T]}^{c}} $ $ [S]=k{{[E]}^{a}}\times {{[v]}^{b}}\times {{[T]}^{c}} $ Putting the dimensions of both sides, we have $ [M{{T}^{-2}}]={{[M{{L}^{2}}{{T}^{-2}}]}^{a}}\times {{[L{{T}^{-1}}]}^{b}}\times {{[T]}^{c}} $ $ [M{{L}^{0}}{{T}^{-2}}]={{[M]}^{a}}\times {{[L]}^{2a+b}}\times {{[T]}^{-2a-b+c}} $ Comparing the powers of both sides, we have $ a=1 $ $ 2a+b=0 $ $ -2a-b+c=-2 $ So, we get a=1, b = - 2 and c = - 2 Hence the dimensional formula for surface tension is, $ T=[E{{v}^{-2}}{{T}^{-2}}] $