The time period T of an oscillating liquid drop is assumed to depend on its radius R , density , and surface tension S . If the percentage errors in the measurement of R , , and S are 2 % , 1 % , and 3 % respectively, what is the maximum percentage error in the calculated value of T ?

Step-by-step solution

Let T = k S^ ^ R^ , where k is a dimensionless constant. Writing the dimensions of each quantity: [T] = [M^0 L^0 T^1] [S] = [M^1 L^0 T⁻²] [ ] = [M^1 L⁻³ T^0] [R] = [M^0 L^1 T^0] Substituting these into the equation: [M^0 L^0 T^1] = [M T⁻²]^ [M L⁻³]^ [L]^ [M^0 L^0 T^1] = [M^ + L^ -3 + T^ -2 ] Equating the powers of M, L , and T on both sides: For T : -2 = 1 = - 1 2 For M : + = 0 = - = 1 2 For L : -3 + = 0 = 3 = 3 2 Therefore, the relation is T S^ -1/2 ^ 1/2 R^ 3/2 . The maximum percentage error in T is given by: T T