Chack the correctness of the relations. (i) escape velocity, `upsilon = sqrt((2GM)/(R ))` (ii) `v =(1)/(2l) sqrt((T)/(m))`, where l is length, T is tension and m is mass per unit length of the string.

Answer

(i) correct (ii) correct

Explanation

(i) `upsilon = sqrt((2GM)/(R ))`
`LHS = upsilon = [M^0L^1T^(-1)]`
`RHS = sqrt((2GM)/(R )) = sqrt((M^(-1)L^3T^(-2)(M))/(L))`
`= sqrtL^2T^(-2) =[M^0L^1T^(-1)]`
AS LHS = RHS, dimensionally, therefore the
formula is correct.
(ii) `v =(1)/(2I)sqrt((T)/(m))`
`LHS = v=[M^0L^0T^(-1)]`
`RHS = (1)/(2I) sqrt((T)/(m)) = (1)/(L) sqrt((MLT^(-2))/(ML^(-1))`
`=(1)/(L)sqrt(L^2T^(-2)) = T^(-1) = [M^0L^0T^(-1)]`
As LHS = RHS, dimensionally, therefore the
formula is correct.