A uniformly tapering conical wire is made fro | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the upper end be at $x=0$ and the lower end at $x=L$. The radius $r(x)$ at a distance $x$ from the top is given by $r(x) = R + \frac{3R-R}{L}x

Vedclass · Accepted Answer

$L\left( {1 + \frac{1}{3}\frac{{Mg}}{{\pi Y{R^2}}}} \right)$

Vedclass · Answer

(C) Let the upper end be at $x=0$ and the lower end at $x=L$. The radius $r(x)$ at a distance $x$ from the top is given by $r(x) = R + \frac{3R-R}{L}x = R(1 + \frac{2x}{L})$.The extension $d\Delta L$ of a small element of length $dx$ is given by $d\Delta L = \frac{Mg dx}{Y A(x)}$,where $A(x) = \pi r(x)^2 = \pi R^2 (1 + \frac{2x}{L})^2$.Integrating from $x=0$ to $x=L$:$\Delta L = \int_0^L \frac{Mg dx}{\pi Y R^2 (1 + \frac{2x}{L})^2} = \frac{Mg}{\pi Y R^2} \int_0^L (1 + \frac{2x}{L})^{-2} dx$.Let $u = 1 + \frac{2x}{L}$,then $du = \frac{2}{L} dx$,or $dx = \frac{L}{2} du$.When $x=0, u=1$; when $x=L, u=3$.$\Delta L = \frac{Mg}{\pi Y R^2} \cdot \frac{L}{2} \int_1^3 u^{-2} du = \frac{MgL}{2 \pi Y R^2} \left[ -\frac{1}{u} \right]_1^3 = \frac{MgL}{2 \pi Y R^2} \left( 1 - \frac{1}{3} \right) = \frac{MgL}{2 \pi Y R^2} \cdot \frac{2}{3} = \frac{MgL}{3 \pi Y R^2}$.The total extended length is $L + \Delta L = L + \frac{MgL}{3 \pi Y R^2} = L \left( 1 + \frac{1}{3} \frac{Mg}{\pi Y R^2} \right)$.

Similar Questions

The value of Young's modulus for a perfectly rigid body is ...........

The following four wires are made of the same material. If the same tension is applied to each,the wire having the largest extension is

Which one of the following is not a unit of Young's modulus?

$A$ truck is pulling a car out of a ditch by means of a steel cable that is $9.1 \, m$ long and has a radius of $5 \, mm$. When the car just begins to move,the tension in the cable is $800 \, N$. How much has the cable stretched? (Young's modulus for steel is $2 \times 10^{11} \, N/m^2$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module