Question

A sonometer consists of two wires of lengths 1.5m and 1m made up of different materials whose densities are 5g/cc and 8g/cc, and their respective radii are in the ratio 4:3. The ratio of tensions in these two wires, if their fundamental frequencies are equal is
A 5 : 3
B 5 : 2
C 2 : 5
D 3 : 5

Answer 1

To solve this question, we must know the formula for the fundamental frequency of wire with both the ends fixed. The frequency depends on the length, tension in the wire and mass per unit of the wire. However, the tension and mass per unit length are the same for all three segments.

Complete step by step answer:
A sonometer wire is a stretched wire. When the wire is vibrated at a point, it undergoes oscillations. These oscillations are called harmonics. The rest harmonic is when the wave has two nodes and only antinodes. Nodes are that points on the wires that have minimum amplitudes and antinodes are those points that have the maximum amplitudes.
We will now compare the fundamental frequencies and will write the expression for it.
$f = F$
Here f and F are the fundamental frequencies whose densities are 5g/cc (first material) and 8g/cc(second material) respectively.
We will further solve the above expression and we get,
$\dfrac{t}{{d{r^2}l}} = \dfrac{T}{{D{R^2}L}}$
Here t,T are the tensions, d and D are the densities, l and L are the lengths and r, R are the radii of the first and second material respectively.
We will now simplify the above equation and we will get,

\Rightarrow \dfrac{t}{T} = \dfrac{{d{r^2}{l^2}}}{{D{R^2}{L^2}}}\\\ \Rightarrow \dfrac{t}{T} = \left( {\dfrac{5}{8}} \right) \times {\left( {\dfrac{4}{3}} \right)^2} \times {\left( {\dfrac{{1.5}}{1}} \right)^2}\\\ \Rightarrow \dfrac{t}{T} = \dfrac{5}{2} \end{array}$$ **Therefore, the correct option is (B).** **Additional information:** Frequency is defined as the number of oscillations or occurrence per unit time. The unit of frequency is Hertz. Frequency is also defined as the reciprocal of time period. From the above calculation you can see that by increasing the length and diameter of the cross-section of wire the frequency decreases. **Note:** When we pluck a string in a sonometer the wire vibrates and produces sound. The plucked wire has vibrations that form a wave that goes to and fro between the ends of the wire. The frequency of the sound wave produced is the same as the frequency of the wave that is produced in the plucked wire.