A piano string `1.5 m` long is made of steel of density `7.7 xx 10^(3) kg//m^(3) and gamma = 2 xx 10^(11) N//m^(2)`. It is maintained at a tension which produces an elastic strain of `1 %` in the string . What is the fundamental frequency of transverse vibration of the string ?

Explanation

Stress `= "gamma" xx "strain" = gamma xx (1)/( 100) = ( 2 xx 10^(11))/(100) = 2 xx 10^(9)`
`T( tension) = pi r^(2) xx stress = pi r^(2) xx 2 xx 10^(9)`
`:. N_(0) = (1)/( 2I) sqrt((I)/(m)) = (1)/( 2 xx 1.5) sqrt ((pi r ^(2) xx 2 xx 10^(9))/( pi r^(2) xx 7700))`
`= 170 Hz`