Question

Two parts of a sonometer wire divided by a movable bridge differ in length by 0.2 cm and produce one beat per second when sounded together. The total length of wire is one metre, then the frequencies are 100 cm
1) 250.5 and 249.5 Hz
2) 230.5 and 229.5 Hz
3) 220.5 and 219.5 Hz
4) 210.5 and 209.5 Hz

Answer 1

Hint: Pythagoras developed the first monochord in the seventh century. Later in the mid-1800s, French instrument maker Albert Marloye transformed the monochord into the differing sonometer. Sonometer is the device used to show the relationship between the length, mass, and tension per unit length of a string and its sound frequency when pressed.

Complete answer:
Total length of the sonometer is 1meter or 100cm. Two parts of the sonometer differ by 0.2 cm. Let us assume, ${{l}_{1}}$ be the length of one part of the wire and ${{l}_{2}}$ be the length of the other part of the wire.

According to the given question,
${{l}_{1}}+{{l}_{2}}=100cm$
And ${{l}_{1}}-{{l}_{2}}=0.2cm$

So, from the above observations, we can derive the value of ${{l}_{1}}$and ${{l}_{2}}$
$l{}_{1}=50.1$ And ${{l}_{2}}=49.9$

We know that, $nl$will remain constant throughout
Therefore the value of $n{{l}_{1}}=(n+1){{l}_{2}}$, now substituting the value of ${{l}_{1}}$and${{l}_{2}}$ in the given equation:
$n\times 50.1=(n+1)49.9$
$n=249.5$
Therefore, ${{n}_{1}}=249.5$
${{n}_{2}}={{n}_{1}}+1$,

By putting value of ${{n}_{1}}$in this equation will give us the value of ${{n}_{2}}$
${{n}_{2}}=249.5+1$
${{n}_{2}}=250.5$
So, for the given question the value of ${{n}_{1}}$and ${{n}_{2}}$ are 249.5 and 250.5 respectively.

Therefore, option (1) is the correct answer for the given question.

Note: For solving these kind of questions we need clear understanding about the instrument that is sonometer. In addition to this, we need to take special care while solving the question to get to the desired outcome.