Question

A column of air and a tuning fork produced \[4\]beats/s when sounded together. The tuning fork gives the lower node. The temperature of the air is \[{15^ \circ }C\]. When the temperature falls to \[{10^ \circ }C\], the two produce three beats per second. Find the frequency of the fork.
\[(A)110Hz\]
\[(B)210Hz\]
\[(C)310Hz\]
\[(D)410Hz\]

Answer 1

Hint: When the temperature of the air is \[{15^ \circ }C\] then a column of air and a tuning fork produces four beats per second. Similarly, when the temperature falls to \[{10^ \circ }C\] then the two produce three beats per second. When the frequency of the temperature decreases at the same time-frequency of the beats also decreases. Now we can find the frequency of the fork using the given logic.

Complete step by step answer:
Tuning fork:
The fork has a handle and two tines. The tines begin to vibrate when the tuning fork is hit with a rubber hammer. The disturbances of surrounding air molecules are produced by the back and forth vibration of the tines. An alternating pattern of high and low-pressure regions is created as the tines continue to vibrate. After being struck on the heel of the hand the tuning fork vibrates at a set of frequencies and is used to assess vibratory sensation and hearing.
To find:
If \[f\] be the frequency of the fork, then the frequency of air column,
\[ = f \pm 4\]
\[ = \dfrac{v}{{4L}}\]
As with a decrease in temperature that beats frequency also decreases, so
\[f + 4 = \dfrac{{{v_{15}}}}{{4L}}\]
Also
\[f + 3 = \dfrac{{{v_{10}}}}{{4L}}\]
Therefore,
\[\dfrac{{f + 4}}{{f + 3}} = \dfrac{{{v_{15}}}}{{{v_{10}}}}\]
\[ = \dfrac{{\sqrt {273 + 15} }}{{\sqrt {273 + 10} }}\]
\[ = 110Hz\]
Therefore,\[f = 110Hz\].
Hence, the correct answer is the frequency of the fork is \[110Hz\].

Note:
A fork-shaped acoustic resonator used in many applications to produce a fixed tone is called the tuning fork.
A U-shaped bar of elastic metal is used to form the prongs. This bar of metal can move freely.
The fork resonates at a specific constant pitch when set vibrating by striking it against an object and also emits a pure musical tone once the high overtones fade out.