A small steel ball of radius r is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity `eta`. After some time the velocity of the ball attains a constant value known as terminal velocity `upsilon_T`. The terminal velocity depends on (i) the mass of the ball m (ii) `eta`, (iii) r and (iv) acceleration due to gravity g . Which of the following relations is dimensionally correct?

Options

A. `upsilon_(T) prop (mg)/(etar)`
B. `upsilon_T prop (etar)/(mg)`
C. `upsilon_(T) prop etarmg`
D. `upsilon_(T) prop (mgr)/eta`

Answer

A

Explanation

`v_T prop m^(a) eta^(b) r^(c) g^d`
`[LT^(-1)] =[M^(a)][ML^(-1)T^(-1)]^b [L]^c [LT^(-2)]^d`
`=[M^(a+b)L^(-b+c+d)T^(-b-2d)]`
`a + b = 0 , a = -b " Let " a = 1, :. b =-1`
and `- b + c +d = 1`
`-b -2d =-1 implies b +2d =1`
`:. d = 1 , c = -1`
Aliter :
To check the dimension of each quantity in R.H.S.
`[(mg)/(etar)]=[LT^(-1)]`