A string is fixed at both ends. The tension in the string and density of the string are accurately known but the length and the radius of cross section of the string are known with some error. If maximum errors made in the measurement of length and radius are 1 % and 0.5 % respectively then what is the maximum possible percentage error in the calculation of fundamental frequency of that string :

Question

A string is fixed at both ends. The tension in the string and density of the string are accurately known but the length and the radius of cross section of the string are known with some error. If maximum errors made in the measurement of length and radius are 1 % and 0.5 % respectively then what is the maximum possible percentage error in the calculation of fundamental frequency of that string : (A) 0.5 % (B) 1.0% (C) 1.5% (D) 2.0%

Tardigrade Admin · Accepted Answer

We use n0=(1/2 l) √(T/μ)=(1/2 l) √(T/ρ s) =(1/2 l) √(T/ρ ⋅ π r2)=(1/2 l r) √(T/ρ π) ⇒ (Δ n0/n)=-(Δ l/l)-(Δ r/r)((Δ f/f)) max =1+0.5=1.5 %

0.5 %

1.0%

1.5%

2.0%

We use n0​=2l1​μT​​=2l1​ρsT​​ =2l1​ρ⋅πr2T​​=2lr1​ρπT​​ ⇒nΔn0​​=−lΔl​−rΔr​(fΔf​)max​ =1+0.5=1.5%

We use
$n_{0} = \frac{1}{2 l} \frac{T}{μ} = \frac{1}{2 l} \frac{T}{ρ s}$
$= \frac{1}{2 l} \frac{T}{ρ \cdot π r ^{2}} = \frac{1}{2 l r} \frac{T}{ρ π}$
$\Rightarrow \frac{Δ n _{0}}{n} = - \frac{Δ l}{l} - \frac{Δ r}{r} (\frac{Δ f}{f})_{m a x}$
$= 1 + 0.5 = 1.5%$