Question

Two $\mathrm{Cu}$ wires of radii $R_1$ and $R_2$, such that $(R_1>R_2)$. Then which of the following is true?

• A. Transverse wave travels faster in thicker wire.
• B. Transverse wave travels faster in thinner wire
• C. Travels with the same speed in both the wires.
• D. Does not travel.

Answer 1

The correct option is B

Transverse wave travels faster in thinner wire

Step 1: Given data

Two $\mathrm{Cu}$ wires of radii $R_1$ and $R_2$, such that $(R_1>R_2)$.

Step 2: Formula used

The speed of transverse wave in a wire,

$v=\sqrt{\frac{T}{\mu }}$

where $T$ is the tension, $\mu$ is the mass per unit length of the wire.

Step 3: Calculation

The speed of transverse wave in a wire,

$v=\sqrt{\frac{T}{\mu }}$

Mass per unit length of the wire,

$$\begin{gathered} \mu =\frac{m}{l} \\ \mu =\frac{m}{V}A \\ \mu =\rho A \end{gathered}$$

were $\rho$ is the density, l is length, $A$ is the area and $V$ is the volume.

Now,

$$\begin{gathered} v=\sqrt{\frac{T}{\mu }} \\ \Rightarrow v=\sqrt{\frac{T}{\rho A}}\left(\because \mu =\rho A\right) \\ \Rightarrow v\propto \frac{1}{\sqrt{A}}\left(\because A={\mathrm{\pi r}}^2\right) \\ \Rightarrow v\propto \frac{1}{\sqrt{{\mathrm{\pi r}}^2}} \\ \Rightarrow v\propto \frac{1}{r} \end{gathered}$$

Since area and radius are inversely proportional, as the radius of the wire increases, the velocity decreases, and when the radius of the wire decreases, velocity increases.

So the transverse waves travel faster in thinner wires.

Option B is correct.