Question

Figure (15-E11) shows a string stretched by a block going over a pulleys. 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 desnity of the material of the block.

Answer 1

Let $\rho \to$ density of the block weight $=\rho Vg$

where V = volume of block

The same tuning fork resonates with string in the two cases.

$f_{10}=\frac{10}{2L}\sqrt{\frac{T}{m}}$

$=\frac{10}{2L}\sqrt{\frac{\rho Vg}{m}}$

($m\to$ mass per unit length of string when the block is immersed in water)

$f_{11}=\frac{11}{2L}\sqrt{\frac{T-{\rho }_{w}Vg}{m}}$

$=\frac{11}{2L}\sqrt{\frac{(\rho -{\rho }_w)Vg}{m}}$

As the f of tuning fork is same

$f_{10}-f_{11}=\frac{10}{2L}\sqrt{\frac{\rho Vg}{m}}$

$=\frac{11}{2L}\sqrt{\frac{(\rho -{\rho }_w)Vg}{m}}$

$\Rightarrow \frac{10}{2L}\sqrt{\frac{\rho -{\rho }_w}{m}}=\frac{\rho -1}{\rho }\frac{100}{121}$

(because $\rho w=1gm/cc$)

$\Rightarrow 100\rho =121\rho -121$

$\Rightarrow \rho =\frac{121}{21}$

$=5.8gm/cc$

$=5.8\times {10}^{3}kg/m^2$