Question

A thin uniform elastic rod of natural length L, density $\rho$ and Young's modulus Y is just completely immersed in vertical position in fluid of density $2\rho$ by applying a vertically downward force at its top end. The rod is in equilibrium, then pick up the correct option. (g is acceleration due to gravity)

• A. There is net compression in rod of magnitude $\frac{\rho L^{2}g}{4Y}$
• B. There is net compression in rod of magnitude $\frac{3}{\rho L^{2}g}2Y$
• C. There is net compression in rod of magnitude $\frac{\rho L^{2}g}{2Y}$
• D. There is net compression in rod of magnitude $\frac{\rho L^{2}g}{Y}$

Answer 1

The correct option is D There is net compression in rod of magnitude $\frac{\rho L^{2}g}{Y}$

$\begin{array}{c}\text{W is weight}F\text{is applied force, V is volume.}\\ \text{Bucyancy force,}B=F+W\end{array}$

$\begin{array}{cc}\Rightarrow F& =B-W\\ & =V\left(2\rho \right)g-V\left(\rho \right)g\\ F& =V\rho g\end{array}$

$\begin{array}{cc}\text{Compression}\Delta L=& \frac{\left(V\rho g\right)L}{AY}\\ =& \frac{V}{A}\frac{\rho gL}{V}(\because V=A\times L)\\ & =\frac{\rho g{L}^2}{Y}\end{array}$