Figure, shows a string stretched by a block going over a pulley. … - Vidyadip

Question

Figure, shows a string stretched by a block going over a pulley. The string vibrates in its tenth harmonic in unison with a particular tuning fork. When a beaker containing water is brought under the block so that the block is completely dipped into the beaker, the string vibrates in its eleventh harmonic. Find the density of the material of the block.

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Answer

Let, $\rho\rightarrow$ density of the block Weight $\rho\text{Vg}$ where V = volume of block The same turning fork resonates with the string in the two cases$\text{f}_{10}=\frac{10}{2\text{l}}\sqrt{\frac{\text{T}-\rho_\text{w}\text{Vg}}{\text{m}}}=\frac{11}{2\text{l}}\sqrt{\frac{(\rho-\rho_\text{w})\text{Vg}}{\text{m}}}$
As the f of tuning fork is same,$\text{f}_{10}=\text{f}_{11}\Rightarrow\frac{10}{2\text{l}}\sqrt{\frac{\rho\text{Vg}}{\text{m}}}=\frac{11}{2\text{l}}\sqrt{\frac{(\rho-\rho_\text{w})\text{Vg}}{\text{m}}}$
$\Rightarrow\frac{10}{11}=\sqrt{\frac{\rho-\rho_\text{w}}{\text{m}}}$
$\Rightarrow\frac{\rho-1}{\rho}=\frac{100}{121}$ $\big($because, $\rho_\text{w}=1\text{gm/cc}\big)$
$\Rightarrow100\rho=121\rho-121$
$\Rightarrow5.8\times10^{3}\text{kg/m}^3$

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