find the rational numbers between -1/2 and 2/3
Answers
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ANSWER :–
new frequency – \begin{gathered}{ \boxed { \bold { {n}_{1} = \frac{n}{4} }}} \\\end{gathered}
n
1
=
4
n
EXPLANATION :–
GIVEN :–
• The fundamental frequency of a sonometer wire is n.
• The length and diameter of the wire are doubled keeping the tension same.
TO FIND :–
New fundamental frequency = ?
SOLUTION :–
• We know that frequency –
\begin{gathered}\\ { \boxed { \bold { n = \frac{1}{2l} \sqrt{ \frac{T}{m} }}}} \\\end{gathered}
n=
2l
1
m
T
• Mass per unit Length :–
m = [ρΠr²(l)]/l
m = ρΠr²
• So that , new equation –
\begin{gathered}\\ { \boxed { \bold { n = \frac{1}{2(2l)} \sqrt{ \frac{T}{ \rho \pi {(2r)}^{2}}}}}} \\\end{gathered}
n=
2(2l)
1
ρπ(2r)
2
T
• Now According to the question –
Length (l) => 2l and radius (r) => 2r
• New fundamental frequency –
\begin{gathered}\\ { \bold { {n}_{1} = \frac{1}{2(2l)} \sqrt{ \frac{T}{ \rho \pi {(2r)}^{2}} }}} \\\end{gathered}
n
1
=
2(2l)
1
ρπ(2r)
2
T
\begin{gathered}\\ { { \bold { {n}_{1} = \frac{1}{4l} \sqrt{ \frac{T}{4 \rho \pi {r}^{2}} }}}} \\\end{gathered}
n
1
=
4l
1
4ρπr
2
T
\begin{gathered}\\ { { \bold { {n}_{1} = \frac{1}{2(4l)} \sqrt{ \frac{T}{ \rho \pi {r}^{2}} }}}} \\\end{gathered}
n
1
=
2(4l)
1
ρπr
2
T
\begin{gathered}\\ { { \bold { {n}_{1} = \frac{1}{4} {n} }}} \\\end{gathered}
n
1
=
4
1
n
\begin{gathered}\\ { { \bold { {n}_{1} = \frac{n}{4} }}} \\\end{gathered}
n
1
=
4
n