Question

The modules, of rigidity, η, the material of a wire, is calculated from the relation

\(\tau\) = \(k.\frac{ηr^4}{2l}\)

where = torsional constant (torque per unit twist), r = radius and l = length.

The relative error, in the determination of η, is calculated to be 0.9%. The relative erros, in the determination \(\tau\) of and l are found to be 0.5% and 0.2%, respectively. The ratio of the least count of the instrument, used to measure r, to the measured (average) value of r, would then equal:

(1) 0.001

(2) 0.0005

(3) 0.002

(4) 0.0001

Answer 1

(2) 0.0005

We have,

\(\tau=k.\frac{ηr^ 4}{2l}\)

∴ η = \(\frac{2}{k}\frac{l\tau}{r^4}\)

∴ \(\frac{Δη}{η}=\frac{Δl}{l}+\frac{Δ\tau}{\tau}+4\frac{Δr}{r}\)

∴ 0.9% = 0.2% + 0.5% + 4 (x)   (x = Δr/r)

This given x(= Δr/r) = 0.05% = 0.0005