The critical velocity v for a liquid depends upon (a) coefficient of viscosity eta (b) density of the liquid rho and © radius of the pipe r. Using dimensions derive an expression for the critical velocity.

The critical velocity v for a liquid depends upon (a) coefficient of v

1.	The critical velocity v for a liquid depends upon (a) coefficient of viscosity eta (b) density of the liquid rho and © radius of the pipe r. Using dimensions derive an expression for the critical velocity.
Answer» Solution :`V prop eta^(x)rho^(y)r^(Z)` `=k eta^(N) rho^(y)r^(z)` .i the dimensional formula of `eta` is `ML^(-1)T^(-1)` TAKING dimensions on both sides `M^(0)LT^(-1)=[ML^(-1)T^(-1)]^(x)[ML^(_3)]^(y)[L]^(z)` `M^(2)LT^(-1)=M^(x+y)L^(-x-3y+z)T^(-x)` Equating the dimensions of T, M nd L on both sides For T, `=1=-x,x=1` For M, `0=x+y,y=-x=-1` for L `1=-x-3y+z` `z=1+x+3y=1+1-3=-1,z=-1` Substiting in equation (i) `v=keta^(1)rho^(-1)r^(-1)=k(eta)/(rho r)`