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A point mass m is suspended at the end of a mass less wire of length l and cross section A. If Y is the Young’s modulus for the wire, obtain the frequency of oscillation for the simple harmonic motion along the vertical line.

A point mass m is suspended at the end of a mass less wire of length l and cross section A. If Y is the Young’s modulus for the wire, obtain the frequency of oscillation for the simple harmonic motion along the vertical line.

Grade:upto college level

1 Answers

Navjyot Kalra
askIITians Faculty 654 Points
6 years ago
From fig. (b), due to equilibrium
T = mg ….(i)
But Y = T /A /ℓ / L
⇒ T = YA ℓ / L
From (i) and (ii)
mg = YA ℓ / L (iii)
From fig. (c)
Restoring force
= - [T - mg] = - [YA (ℓ + x ) / L - YA ℓ / L] [from (iii)]
= - YAx /L
On comparing this equation with F = - mω2x, we get
mω2 = YA / L
⇒ ω = √ YA / mL ⇒ 2π / T = √YA / mL
Frequency f = 1 / T = 1 / 2π √ YA / mL

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