State the principle of homogeneity. Test the dimensional homogeneity of equations - (i) s = ut + 1/2 at^(2) (ii) S_(n) = u + a/2 (2n - 1)

State the principle of homogeneity. Test the dimensional homogeneity o

1.	State the principle of homogeneity. Test the dimensional homogeneity of equations - (i) s = ut + 1/2 at^(2) (ii) S_(n) = u + a/2 (2n - 1)
Answer» Solution :(i) DIMENSION of L.H.S [s] = `[M^(0)L^(1)T^(0)]` Dimension of R.H.S. = `[ut] + [at^(2)]` = `[LT^(-1).T] + [M^(0)L^(1)T^(-2).T] = [M^(0)L^(1)T^(0)]` as Dimension of L.H.S = Dimension of R.H.S `:.` The equation is DIMENSIONALLY homogeneous. (ii) `S_(n) =` Distance TRAVELLED in `n^(th)` sec that is `(S_(n) - S_(n-1))` `:.S_(n) = U xx 1 + a/2 [2N - 1]` `[LT^(-1)] = [LT^(-1)] + [LT^(-2)][T]` `[LT^(-1)] = [LT^(-2)]` L.H.S. = R.H.S. Hence this is dimensionally correct.