Using dimensions show that the viscous force acting on a glass sphere falling through a highly viscous liquid of coefficient of viscosity eta is Fprop eta av where a is the radius of the sphere and v its terminal velocity.

Using dimensions show that the viscous force acting on a glass sphere

1.	Using dimensions show that the viscous force acting on a glass sphere falling through a highly viscous liquid of coefficient of viscosity eta is Fprop eta av where a is the radius of the sphere and v its terminal velocity.
Answer» Solution :Dimensional formula of `eta` is `[ML^(-1)T^(-1)]` `F PROP eta^(x)a^(y)v^(z)` `F=keta^(x)a^(y)v^(z)`, k is dimensionaless constant Taking DIMENSIONS on both SIDES `MLT^(-2)=[ML^(-1)T^(-1)]^(x)[L]^(y)[LT^(-1)]^(z)` `MLT^(-2)=M^(x)L^(-1)T^(-x)L^(y)T^(-z)` `MLT^(-2)=M^(x)L^(-x+y+z)T^(x-x-z)` EQUATING dimensions on both sides of M, `1=x""` i.e. x=1 of T `2=-x-z""-z=-2+x=-2+1,z=1` of L `1=-x+y+z=-1+y+1""` i.e. `y=1` `F=k eta^(1)a^(1)v^(1)` or `F prop eta av`.