Question: A column of air and a tuning fork produced \[4\]beats/s when sounded together. The tuning fork gives...

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

Solution

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.