It has been observed that velocity of ripple waves produced in water `( rho)` , and surface tension `(T)` . Prove that ` V^(2) prop T// lambda rho`.

Answer

`v^(2) prop (T)/ (lambda rho)`

Explanation

According to the problem ,
` v prop lambda ^(a) rho^(b) T^(c )`
` v = k lambda ^(a) rho^(b) T^(c )`
Where `k` is a dimensionless constant .
` LT%(-1) = L^(a) (ML^(-3))^(b) ( MT^(-2))^( c )`
rArr `M^(0) L^(1) T^(-1) = M^( b + c) L^( a - 3b ) T^(-2 c )`
Using the principle of homogenity , we get
` b + c = 0 , a - 3 b = 1 , -2c = -1 `
Solving these equations , we get
`a = -(1)/(2) , b = -(1)/( 2) , c = (1)/(2)`
So , the relation becomes ` v = k lambda ^(-1//2) rho^(-1//2) T^(1//2) rArr v^(2) prop (T)/( lambda rho)`