The period of vibration of a tunign fork depends on the length I of its prong, density d and Young's modulus Y of the meterial. Deduce an expression for the period of vibration (T) using dimensional analysis.

Answer

`T = KI sqrt(d//Y)`

Explanation

Let `T =K l^a d^b Y^c… (i)`
where K is dimensionless constant.
Writing the dimensions in (i), we get
`[M^0 L^0 T^1] =L^a(ML^(-3))^b(ML^(-1) T^(-2))^c`
`=M^(b+c) L^(a-3b-c)T^(-2c)`
Equating the dimensions of M, L, T on both
sides, we get
` b+c= 0, a-3b-c = 0, -2c =1, c= -(1)/(2)`
`b = -c =(1)/(2), a =3b +c =(3)/(2) -(1)/(2) =1`
From (i)
`T =Kl^1 d^(1//2) Y^(-1//2)`
`=K Isqrt((d)/(Y))`