Question
State and verify the laws of vibrating strings using a sonometer.
✓
Answer
i. Law of length: The fundamental frequency of vibrations of a string is inversely proportional to the length of the vibrating string if tension and mass per unit length are constant.
$\therefore n \propto \frac{1}{l}$
(If $T$ and $m$ are constant)
Verification of first law:
a. By measuring the length of wire and its mass, the mass per unit length (m) of wire is determined. Then the wire is stretched on the sonometer and the hanger is suspended from its free end.
b. A suitable tension $(T)$ is applied to the wire by placing slotted weights on the hanger.
c. The length of wire $(l_1)$ vibrating with the same frequency $(n_1)$ as that of the tuning fork is determined as follows.
d. A light paper rider is placed on the wire midway between the bridges. The tuning fork is set into vibrations by striking on a rubber pad.
e. The stem of the tuning fork is held in contact with the sonometer box. By changing the distance between the bridges without disturbing the paper rider, the frequency of vibrations of the wire is changed.
f. When the frequency of vibrations of wire becomes exactly equal to the frequency of tuning fork, the wire vibrates with maximum amplitude, and the paper rider is thrown off.
g. In this way a set of tuning forks having different frequencies $n_1, n_2, n_3, …………$are used and corresponding vibrating lengths of wire are noted as $l_1, l_2, l_3………$.by keeping the tension constant (T).
h. It is observe that $n_1l_1 = n_2l_2 = n_3l_3 =……. =$ constant, for the constant value of tension (T) and mass per unit length (m).
$\therefore nl$= constant
i.e., $n \propto \frac{1}{l}$, if $T$ and $m$ are constant.
Thus, the first law of a vibrating string is verified by using a sonometer.
Law of tension: The fundamental frequency of vibrations of a string is directly proportional to the square root of tension if the vibrating length and mass per unit length are constant.
$\therefore n \propto \sqrt{ T } \ldots$...(If I and $m$ are constant)
Verification of second law:
a. The vibrating length (I) of the given wire of mass per unit length $( m )$ is kept constant for verification of the second law.
b. By changing the tension, the same length is made to vibrate in unison with different tuning forks of various frequencies.
c. If tensions $T_1, T_2, T_3 \ldots \ldots .$. correspond to frequencies $n_1, n_2, n_3, \ldots \ldots .$. etc. It is observed that $\frac{ n _1}{\sqrt{ T _1}}=\frac{ n _2}{\sqrt{ T _2}}=\frac{ n _3}{\sqrt{ T _3}}=\ldots \ldots=$ constant $\therefore \frac{n}{\sqrt{ T }}=$ constant
$\therefore n \propto \sqrt{ T }$ if $I$ and $m$ are constant. Thus, the law of tension of $a$ vibrating string is verified by using a sonometer.
iii. Law of linear density: The fundamental frequency of vibrations of a string is inversely proportional to the square root of mass per unit length (linear density) if the tension and vibrating length of the string are constant.
$\therefore n \propto \sqrt{\frac{1}{ m }}$ ...(If I and T are constant)
Verification of third law:
a. For verification of the third law of a vibrating string, two wires having different masses per unit lengths $m _1$ and $m _2$ (linear densities) are used.
b. The first wire is subjected to suitable tension and made to vibrate in unison with a given tuning fork.
c. The vibrating length is noted as $\left( l _1\right)$. Using the same fork, the second wire is made to vibrate under the same tension and the vibrating length $\left( I _2\right)$ is determined.
d. Thus the frequency of vibration of the two wires is kept the same under the same applied tension $T$. It is found that
$ l_1 \sqrt{m}_1=l_2 \sqrt{m}_2$
$l \sqrt{m}=\text { constant } $
e. But by the first law of a vibrating string, $n \propto \frac{1}{l}$
Therefore, $n \propto \frac{1}{\sqrt{m}}$, if $T$ and I are constant. Thus, the third law of vibrating string is verified by using a sonometer.
$\therefore n \propto \frac{1}{l}$
(If $T$ and $m$ are constant)
Verification of first law:
a. By measuring the length of wire and its mass, the mass per unit length (m) of wire is determined. Then the wire is stretched on the sonometer and the hanger is suspended from its free end.
b. A suitable tension $(T)$ is applied to the wire by placing slotted weights on the hanger.
c. The length of wire $(l_1)$ vibrating with the same frequency $(n_1)$ as that of the tuning fork is determined as follows.
d. A light paper rider is placed on the wire midway between the bridges. The tuning fork is set into vibrations by striking on a rubber pad.
e. The stem of the tuning fork is held in contact with the sonometer box. By changing the distance between the bridges without disturbing the paper rider, the frequency of vibrations of the wire is changed.
f. When the frequency of vibrations of wire becomes exactly equal to the frequency of tuning fork, the wire vibrates with maximum amplitude, and the paper rider is thrown off.
g. In this way a set of tuning forks having different frequencies $n_1, n_2, n_3, …………$are used and corresponding vibrating lengths of wire are noted as $l_1, l_2, l_3………$.by keeping the tension constant (T).
h. It is observe that $n_1l_1 = n_2l_2 = n_3l_3 =……. =$ constant, for the constant value of tension (T) and mass per unit length (m).
$\therefore nl$= constant
i.e., $n \propto \frac{1}{l}$, if $T$ and $m$ are constant.
Thus, the first law of a vibrating string is verified by using a sonometer.
Law of tension: The fundamental frequency of vibrations of a string is directly proportional to the square root of tension if the vibrating length and mass per unit length are constant.
$\therefore n \propto \sqrt{ T } \ldots$...(If I and $m$ are constant)
Verification of second law:
a. The vibrating length (I) of the given wire of mass per unit length $( m )$ is kept constant for verification of the second law.
b. By changing the tension, the same length is made to vibrate in unison with different tuning forks of various frequencies.
c. If tensions $T_1, T_2, T_3 \ldots \ldots .$. correspond to frequencies $n_1, n_2, n_3, \ldots \ldots .$. etc. It is observed that $\frac{ n _1}{\sqrt{ T _1}}=\frac{ n _2}{\sqrt{ T _2}}=\frac{ n _3}{\sqrt{ T _3}}=\ldots \ldots=$ constant $\therefore \frac{n}{\sqrt{ T }}=$ constant
$\therefore n \propto \sqrt{ T }$ if $I$ and $m$ are constant. Thus, the law of tension of $a$ vibrating string is verified by using a sonometer.
iii. Law of linear density: The fundamental frequency of vibrations of a string is inversely proportional to the square root of mass per unit length (linear density) if the tension and vibrating length of the string are constant.
$\therefore n \propto \sqrt{\frac{1}{ m }}$ ...(If I and T are constant)
Verification of third law:
a. For verification of the third law of a vibrating string, two wires having different masses per unit lengths $m _1$ and $m _2$ (linear densities) are used.
b. The first wire is subjected to suitable tension and made to vibrate in unison with a given tuning fork.
c. The vibrating length is noted as $\left( l _1\right)$. Using the same fork, the second wire is made to vibrate under the same tension and the vibrating length $\left( I _2\right)$ is determined.
d. Thus the frequency of vibration of the two wires is kept the same under the same applied tension $T$. It is found that
$ l_1 \sqrt{m}_1=l_2 \sqrt{m}_2$
$l \sqrt{m}=\text { constant } $
e. But by the first law of a vibrating string, $n \propto \frac{1}{l}$
Therefore, $n \propto \frac{1}{\sqrt{m}}$, if $T$ and I are constant. Thus, the third law of vibrating string is verified by using a sonometer.
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