a disc of 10 cm radius and 2 kg mass is suspended is horizontal plane by a vertical wire attached to its center. if the diameter of wire us 1.5 cm and periods of torsional oscillation of disc is 10 sec find rigidly of material of wire.
Answers
Answer ⇒ 2π²mr²/T²
Explanation ⇒ Let the torsional constant be k.
If the torque in the wire be for an angular displacement of θ, then
= kθ
Also, By the definition of Torque, = Iα, where I is the moment of inertia and α is the angular acceleration.
Now, moment of inertia of the disc about the axis passing through its centre = mr²/2
∴ α = /Iα
∴ α = kθ/mr²/2
∴ α = 2kx/mr³ [∵ θ =x/r]
∴ a/r = (2k/mr³)x [∵ α = a/r]
∴ a = (2k/mr²)x
Since the acceleration is proportional to the displacement, this means the motion for this tortion is an SHM.
Comparing it with a = ω²x
∴ ⍵ = √(2k/mr²)
Now, time period =2π/⍵
∴ T = 2π√(mr²/2k)
∴ k = 2π²mr²/T²
Hence, the torsional constant of the wire is 2π²mr²/T².
Hope it helps.
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