Question

​​​​​​The mass m, of a stone, that can be moved by flowing stream of water, depends on the velocity (v), of the flowing water, the density (p) of water and the acceleration due to gravity (g). The relation, between the different quantities, is expressed as

m = kvⁿ pg^-3

The relative percentage errors, in the determination of v, p and g are 1%, 0.7% and 0.1%, respectively. The value of n, in the above relation, and the relative percentage error, in the determination of m, are equal, respectively, to

(1) 5; 6%

(2) 6; 5%

(3) 7; 6%

(4) 6; 7%

Answer 1

(4) 6; 7%

Using the principle of dimensional homogenity, in the relation,

m = kvⁿ pg^-3 , we get 

[M] = [LT^-1]ⁿ [ML^-3] [LT^-2]^-3 

We get – n + 6 = 0 or n = 6

This value is also consistent with n + (-3) + (-3) = 0

∴ m = kv⁶ pg^-3

∴ \(\frac{Δm}{m}=6\frac{Δv}{v}+\frac{Δp}{p}+3\frac{Δg}{g}\)

= 6 (1%) + (0.7%) + 3(0.1%) 

= 7%