Answer to Question #126271 in Mechanics | Relativity for Che

Question #126271
A rod, 1 cm2 x 200 cm and mass 2 kg, is clamped at its center. When vibrating longitudinally it emits its third overtone in unison with a tuning fork making 6000 vps. How much will the rod be elongated if, when clamped at one end, a stretching force of 980 N is applied to the other end?
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Expert's answer
2020-07-14T08:51:41-0400

Explanations & Calculations

  • Velocity of the longitudinal waves through a dense medium = Eρ\sqrt {\frac{E}{\rho}}
  • During the 3rd overtone the length of the rod (ll )= 7λ2\frac{7\lambda}{2} when the rod is clamped at its center. therefore, the frequency of the third overtone

f=vλ=7v2l=72lEρ\qquad\qquad \begin{aligned} \small f &= \small \frac{v}{\lambda}= \frac{7v}{2l} = \frac{7}{2l}\sqrt\frac{E}{\rho} \end{aligned}

  • ρ=mvolume=2kg200×106m3=10000kgm3\small \rho = \large \frac{m}{volume } = \large\frac{2kg}{200\times10^{-6}m^3} = \small 10000kgm^{-3}


  • Since the rod is in resonance with the tuning fork, f =6000
  • Therefore,

6000=72×2mE10000E=1.17551×1011Nm2\qquad\qquad \begin{aligned} \small 6000 & = \small \frac{7}{2\times 2m}\sqrt{\frac{E}{10000}}\\ \small E &= \small 1.17551\times 10^{11}Nm^{-2} \end{aligned}


  • Therefore, the elongation (e) could be calculated to be,

FA=Eele=FlAE=980N×2m104m2×(1.17551×1011Nm2)=1.667×104m=0.1667mm\qquad\qquad \begin{aligned} \small \frac{F}{A} & =\small E \frac{e}{l}\\ \small e & = \small \frac{F l}{AE}\\ &= \small \frac{980N\times2m}{10^{-4}m^2\times (1.17551\times 10^{11}Nm^{-2})}\\ &= \small \bold{1.667\times 10^{-4}m}\\ &= \small \bold{0.1667mm} \end{aligned}


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