Question

Assuming that the period of vibration of a tuning fork depends upon the length of the prongs 1/).onthe density (d) and the Young's modulus of the material (Y), find by the method of dimensions, theformula for the period of vibration.​

Answer 1

Given,

Assuming that the period of vibration of a tuning fork depends upon the length of the prongs 1/).onthe density (d) and the Young's modulus of the material (Y).

To find,

the formula of period vibration by the method of dimensions.

• dimension of length, l = [L]
• dimension of density, d = [ML¯³]
• dimension of Young's modulus, Y = [ML¯¹T¯²]
• dimension of time period, T = [T]

let T ∝ l^x × d^y × Y^z

from dimensional analysis,

⇒[T] ∝ [L]^x × [ML¯³]^y × [ML¯¹T¯²]^z

⇒[T] ∝ [M^(y + z) L^(x - 3y - z) T^(-2z)]

on comparing we get,

y + z = 0, x - 3y - z = 0, -2z = 1

x =1 , y = 1/2 , z = -1/2

so, T ∝ l^1 d^1/2 Y^-1/2

therefore, time period of vibration, T = kl√{d/Y} , where k is proportionality constant.

Answer 2

T = kl√p/y

Explanation:

T depends on l, p and Y

T = k * l^a * p^b * y^c where k is the dimensional constant of the tuning fork.

M°L°T = k [M°L T°]^a [M L^-3 T°]^b [M L^-1 T^-2]^c when we substitute the dimension of each quantity

M°L°T  = k [ M^b+c. L^e-3b-c. T^-2c]

Equating the powers of M L and T, we get:

b+c = 0

b = -c

a - 2b - c = 0

a = 2b = 1

- 2c = 1

c = -1/2

So b = 1/2

Substituting the values of a, b, and c, we get:

T = k. l. p^1/2. y^-1/2

Therefore T = kl√p/y