Question

A 20 cm wide thin circular disc of mass 200 g is suspended to a rigid support from a thin metallic string. By holding the rim of the disc, the string is twisted through 60^o and released. It now performs angular oscillations of period 1 second. Calculate the maximum restoring torque generated in the string under undamped conditions. (π³ ≈ 31)

Answer 1

Given: R = 10cm = 0.1 m, M = 0.2 kg, θ_m=60⁰ = p/3 rad, T=1 s, π³ = 31

The MI of the disc about the rotation axis (perpendicular through its centre) is

I = ½ MR² = ½ (0.2)(0.1)² = 10^-3 kg-m²

The period of torsional oscillation, T = 2π\(\sqrt{\frac{I}{c}}\)

∴ The torsion constant, c = 4π²\(\frac{I}{T^2}\)

The magnitude of the maximum restoring torque,

τ_max = cθ_m = 4π²\(\frac{I}{T^2}(\frac{π}{3})\) 

= \(\frac{4}{3}π^3\frac{I}{T^2}=\frac{4}{3}(31)(\frac{10^{-3}}{1^2})\) 

= 41.33 x 10^-3 = 0.04133 N-m