In a sonometer experiment the bridges are separted by a fixed distance the wire which is slightly elastic emits a tone of frequency n when held by tension T If the tension is increased to 4T the tone emitted by the wire will be of frequency

Options

A. f
B. 2f
C. Slightly greater than 2f
D. Slightly less than 2f

Answer

D

Explanation

Since frequency `(f)=(1)/(2l)sqrt((T)/(mu))`
: `mu rarr` Mass per unit length
`mu=("Mass")/("Length")=((Al)rho)/(l)=Arho`
`mu=(pir^(2))rho`
`f=(1)/(2l)sqrt((T)/((pir^(2))rho))...(i)`
If tension is increased, the wire gets elongated
`f.=(1)/(2l.)sqrt((4T)/(pi(r.)^(2)rho))...(ii)`
From eq. (i) & (ii)
`(f.)/(f)=(2l)/(l.)xx(r)/(r.)`
Since, mass remains constant
`(pir^(2))lrho=pi(r.)^(2)l.rhoimplies(r)/(r.)=sqrt((l.)/(l))`
`therefore (f.)/(f)=2sqrt((l.)/(l)).(l)/(l.)=2sqrt((l)/(l.))`
As `l.gtl` hence `f.lt2f`.