A sonometer wire under tension of 63 N vibrating in its fundamental mode is in resonance with a vibrating tuning fork. The vibrating portion of that sonometer wire has a length of 10 cm and a mass of 1 g. The vibrating tuning fork is now moved away from the vibrating wire with a constant speed and an observer standing near the sonometer hears one beat per second. Calculate the speed with which the tuning fork is moved if the speed of sound in air is 300 m/s.

Explanation

The fundamental frequency of wire is equal to the frequency of the stationary fork, which is given by
`n=(1)/(2l)sqrt(((T)/(m)))`
Here `m=(1g)/(10cm)=(1xx10^-3kg)/(10xx10^-2m)=10^-2(kg)/(m)`,
`T=64N`,`l=10cm=0.1m`
`n=(1)/(2xx0.1)xxsqrt(((64)/(10^-2)))=400Hz`
when the fork moves away from the source its apparent frequency decreses. As the number of beats heard per second is 1, apparent frequency of moving fork is `400-1=399`. The apparent frequency due to recending source is given by
`n='=(v)/(v+v_S)n`
or `399=(300)/(300+v_S)xx400`
or `399(300+v_S)=300xx400`
or `399v_S=300`
or `v_S=(300)/(399)=0.752(m)/(s)`