First slide

Elastic Modulie

Question

The elongation in a metallic rod hinged at one end and rotating in a horizontal plane becomes four times of the initial value. The angular velocity of rotation becomes

Moderate

A

two times the initial value
B

half of initial value
C

one third of initial value
D

four times the initial value

Solution

$- \int_{T}^{0} ∆ T = \int_{0}^{l} \frac{m}{l} {dxω}^{2} x \Rightarrow T = \frac{m}{l} ω^{2} \frac{x^{2}}{2}$
$\Rightarrow Y = \frac{Fl}{A ∆ l}$
$δl = \frac{Fl}{AY}$
$δl = \frac{\frac{m}{l} \frac{ω^{2} x^{2}}{2} dx}{AY}$
$∆ l = \int δl = \frac{{mω}^{2} l^{3}}{6 AYl}$
$∆ l = \frac{{ρω}^{2} l^{3}}{6 Y}$
$∆ l \propto ω^{2}$
For elongation to become 4 times the final angular speed must be
$ω_{2} = 2 ω_{1}$

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