Question

A dimensionless number (N_R), called Reynold’s number, can be made from a suitable combination of r (radius of the tube), p (density of the liquid), v (velocity of the fluid) and η (the coefficient of viscosity of the liquid). The relative percentage errors, in the measurement of r, p , v and η are found to be 0.4%, 1.5%, 2% and 2.5% respectively. 

The formula, for NR , and the relative percentage error in its calculated value, are equal, respectively, to:

(1) N_R = \(\frac{rpv}{η};\) \(\frac{ΔN_R}{N_R} = 6.4\) %

(2) N_R = \(\frac{rpv}{η};\) \(\frac{ΔN_R}{N_R} = 2.4\) %

(3) N_R = \(\frac{rpv}{η};\) \(\frac{ΔN_R}{N_R} = 1.4\) %

(4) N_R = \(\frac{rpv}{η};\) \(\frac{ΔN_R}{N_R} = 6.2\) %

Answer 1

(1) N_R = \(\frac{rpv}{η};\) \(\frac{ΔN_R}{N_R} = 6.4\) %

We know that 

[r] = [L]; [p] = [ML^-3]

[v] = [LT^-1]; η = [ML^-1 T^-1]

These give [rpv/η] as a dimensionaless combination.

Thus N_R = rpv/η

∴ \(\frac{ΔN_R}{N_R}=\frac{Δr}{r}+\frac{Δv}{v}+\frac{Δη}{η}\)

= 0.4% + 1.5% + 2% + 2.5% = 6.4%