the frequency of an open organ pipe at 15°C is 156Hz. At what temperature will frequency be 160Hz?
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Explanation:
Fundamental frequency of an organ pipe,
f=
2l
v
where l is length of the organ pipe at temperature T
Also, l=l
0
[1+α(T−T
0
)]
Velocity of sound, v=
M
γRT
we have to find the temperature at which f(T)=f(T
0
) for small (T−T
0
)
2l
0
[1+α(T−T
0
)]
M
γRT
=
2l
0
M
γRT
⟹
T
0
T
=1+α(T−T
0
)
[1+(
T
0
T−T
0
)]
1/2
=1+α(T−T
0
)
For small (T−T
0
) we may use binomial approximation,
1+
2
1
(
T
0
T−T
0
)=1+α(T−T
0
)
2
1
(
T
0
T−T
0
)=α(T−T
0
),α=
2T
0
1
or T
0
=
2α
1
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