Question

Consider a rope of total mass $M$ and length $L$ suspended at rest from a fixed mount. The rope has a linear mass density that varies with height as $\lambda \left(z\right)={\lambda }_0\sin (\pi z\setminus L)$, where ${\lambda }_0$ is a constant. Constant gravitational acceleration $g$ acts downward, then constant ${\lambda }_0$ is

• A. $\frac{\pi M}{2L}$
• B. $\frac{\pi M}{3L}$
• C. $\frac{\pi M}{4L}$
• D. $\frac{2\pi M}{L}$

Answer 1

The correct option is A $\frac{\pi M}{2L}$

Consider the rod as

We consider length of the rod from bottom to top.
We have been given that
$\lambda ={\lambda }_0\sin \left(\frac{\pi z}{L}\right)$
$\Rightarrow \frac{dm}{dz}={\lambda }_0\sin \left(\frac{\pi z}{L}\right)$
$\Rightarrow {\int }_0^{M}dm={\int }_0^L{\lambda }_0\sin \left(\frac{\pi z}{L}\right)dz$
$\Rightarrow M=\frac{{\lambda }_{0}L}{\pi }[-\cos \left(\frac{\pi z}{L}\right){]}_0^L$
$\Rightarrow M=\frac{{\lambda }_{0}L}{\pi }[1+1]$
$\Rightarrow {\lambda }_0=\frac{\pi M}{2L}$