If velocity, force and time are taken as the fundamental quantities, then using dimensional analysis choose the correct dimensional formula for mass among the following. [ K is a dimensionless constant ]

Question

If velocity, force and time are taken as the fundamental quantities, then using dimensional analysis choose the correct dimensional formula for mass among the following. [ K is a dimensionless constant ], Q = K v − 1 F T, Q = K v 3 F T 2, Q = 2 K v − 1 F T, Q = 3 K v 2 F T

MARKS App · Accepted Answer

Let the quantity be Q, then, Q = f ( v , F , T ) Assuming that the function is the product of power functions of v , F and T , Q = K v x F y T z . . . ( i ) where K is a dimensionless constant of proportionality. The above equation dimensionally becomes [ Q ] = [ L T - 1 ] x [ M L T - 2 ] y [ T ] z i.e., [ Q ] = [ M y L ( x + y ) T ( - x - 2 y + z ) ] , ​Now Q = mass i.e., [ Q ] = [ M ] So Equation (ii) becomes [ M ] = [ M y L ( x + y ) T ( - x - 2 y + z ) ] its dimensional correctness requires y = 1 , x + y = 0 and - x - 2 y + z = 0 which on solving yields x = - 1 , y = 1 and z = 1 Substituting it in Equation (i), we get Q = K v - 1 F T

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