First slide

Elastic behaviour of solids

Question

A thin uniform metallic rod of length 0.5 m and radius 0.1 m with an angular velocity 400 rad s-¹ in a horizontal plane about a vertical axis passing through one of its ends. Elongation in the rod is (in m) [Given density of material of the rod is $10^{4} {kgm}^{- 3} and Y = 2 \times 10^{11} {Nm}^{- 2}]$

Moderate

A

$\frac{10^{- 3}}{3}$
B

$\frac{3}{10^{- 3}}$
C

$\frac{2}{10^{2}}$
D

$\frac{10^{2}}{2}$

Solution

Consider an element of length dx at a distance x from the axis of rotation,
$dF = {dmxω}^{2} = ρAdx \cdot x \cdot ω^{2} F = {ρAω}^{2} \int_{x}^{L} xdx = \frac{1}{2} {ρAω}^{2} (L^{2} - x^{2})$
If dy is the elongation in the element of length dx, then
$\frac{dy}{dx} = \frac{F}{AY} dy = {\frac{Fdx}{A \cdot Y}|}_{Δl} = \frac{{ρω}^{2}}{2 Y} \int_{0}^{L} (L^{2} - x^{2}) dx Δl = \frac{{ρω}^{2} \cdot L^{2}}{3 Y} Δl = \frac{10^{4} \times (400)^{2} \times {(\frac{1}{2})}^{3}}{2 \times 10^{11}} = \frac{10^{- 3}}{3} m$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App