A string is stretched by a block going over a pulley . The string vibrates in its fifth harmonic in unison with a particular tuning fork . When a beaker containing a liquid of density `rho` is brought under the block so that the block is completely dipped into the beaker , the string vibrates in its seventh harmonic in unison with the tuning fork . Find the density of the material of the block.

Answer

`(49)/(24) rho`

Explanation

Let `sigma` be the density of the block
` n = ( 5 v_(1))/( 2 L) = ( 7 v_(2))/( 2L)` (i)
`5 v_(1) = 7 v_(2)` (ii)
`5 sqrt(T_(1)) = 7 sqrt(T_(2))`
where `T_(1) = V sigma g and T_(2) = V sigma g - V rho g`
Squaring both sides of Eq. (ii)
`25 T_(1) = 49 T_(2)`
`25 V sigma rho = 49 (V sigma g - V rho g)`
`( 49 - 25) V sigma g = 49 V rho g`
`sigma = ( 49)/(24) rho`