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.
Answers
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.
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