Two wires of the same material are vibrating | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The frequency of the $p$-th overtone for a string fixed at both ends is given by $f = (p+1) \frac{1}{2 \ell} \sqrt{\frac{T}{\mu}}$,where $\mu = \p

Vedclass · Accepted Answer

$1 : 3$

Vedclass · Answer

(A) The frequency of the $p$-th overtone for a string fixed at both ends is given by $f = (p+1) \frac{1}{2 \ell} \sqrt{\frac{T}{\mu}}$,where $\mu = \pi r^2 \rho$.Since the material is the same,density $\rho$ is constant. The tension $T$ is also constant.For the first wire,the first overtone $(p=1)$ is $f_1 = 2 \cdot \frac{1}{2 \ell_1} \sqrt{\frac{T}{\pi r_1^2 \rho}} = \frac{1}{\ell_1 r_1} \sqrt{\frac{T}{\pi \rho}}$.For the second wire,the second overtone $(p=2)$ is $f_2 = 3 \cdot \frac{1}{2 \ell_2} \sqrt{\frac{T}{\pi r_2^2 \rho}} = \frac{3}{2 \ell_2 r_2} \sqrt{\frac{T}{\pi \rho}}$.Given $f_1 = f_2$,we have $\frac{1}{\ell_1 r_1} = \frac{3}{2 \ell_2 r_2}$.Given $r_1 = 2 r_2$,substituting this into the equation: $\frac{1}{\ell_1 (2 r_2)} = \frac{3}{2 \ell_2 r_2}$.Simplifying,$\frac{1}{2 \ell_1} = \frac{3}{2 \ell_2}$,which gives $\frac{\ell_1}{\ell_2} = \frac{1}{3}$.

Two wires of the same material are vibrating under the same tension. If the first overtone of the first wire is equal to the second overtone of the second wire and the radius of the first wire is twice the radius of the second,then the ratio of the length of the first wire to the second wire is:

Similar Questions

$A$ tuning fork vibrating with a sonometer having $20 \ cm$ wire produces $5$ beats per second. The beat frequency does not change if the length of the wire is changed to $21 \ cm$. The frequency of the tuning fork (in Hertz) must be

$A$ sonometer wire of length $25 \,cm$ vibrates in unison with a tuning fork. When its length is decreased by $1 \,cm$, $6$ beats are heard per second. What is the frequency of the tuning fork (in $\,Hz$)?

Two strings of the same material having lengths $L$ and $2L$,and radii $2r$ and $r$ respectively,are vibrating in the fundamental mode. The tension applied to both strings is the same. The ratio of their respective fundamental frequencies is:

$A$ uniform string of length $L$ and mass $M$ is fixed at both ends while it is subject to a tension $T$. It can vibrate at frequencies $(v)$ given by the formula (where $n=1, 2, 3, \ldots$):

Vedclass Test Series

Exam Paper Generator

Online Exam Module