Clamped at the middle a metal rod of length 1 meter and density 7.5gm/cc,gives dust heaps at intervals of 8cm" velocity of sound in the gas used is 400 meter/sec.then the young's modulus of material of the rod will be

Explanation

The situation is shown in figure-6.75.

Let `lambda_(r)` and `lambda_(g)` be the wavelength in the rod and gas column respectively. Since the rod is clampted in the middle, it vibrates with a node in the middle and antinodes at the ends. Therefore
`lambda_(r)/(2) = 1m = 100 cm`
`lambda_(g)/(2) = 8 cm` If `v_(r)` and `v_(g)` be the sound velocities in the rod and in the gas respectively and n be the frequency of vibration, then frequency of oscillations is given as
`n = (v_(rod))/(lambda_(rod)) = (v_(gas))/(lambda_(gas))`
`v_(rod) = (lambda_(rod))/(lambda_(gas)) xx v_(gas)`
= `(200)/(16) xx 400`
= 5000 m/s
Now velocityof longitudinal waves in a metal rod is given by
`v_(rod) = sqrt(((Y)/d))`
`Y = v_(rod)^(2) xx d`
`Y = (500.0)^(2) xx 7.5 xx 10^(3)`
= `1.875 xx 10^(11)N//m^(2)`