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Question
A 20 cm wide thin circular disc of mass 200 g is suspended to rigid support from a thin metallic string. By holding the rim of the disc, the string is twisted through 60° 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)
Solution
Data: R = 10 cm = 0.1 m, M = 0.2 kg, θm = 60° = `"π"/3`rad, T = 1 s, π3 ≈ 31
The MI of the disc about the rotation axis (perperdicular through its centre) is
I = `1/2`MR2 = `1/2`(0.2)(0.1)2 = 10−3 kg.m2
The period of torsional oscillation, T = `2πsqrt("I"/"c")`
∴ The torsion constant, c = `4π^2"I"/"T"^2`
The magnitude of the maximum restoring torque,
`τ_"max"` = cθm = `(4π^2"I"/"T"^2)(π/3)`
= `4/3π^3"I"/"T"^2=4/3(31)((10^-3)/1^2)`
= 41.33 × 10−3 = 0.04133 N.m
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