Question 453: Waves

Question

The extension in a wire obeying Hooke's law is $$x$$. The speed of sound in the stretched wire is $$v$$. If the extension in the wire is increased to $$4 x$$, then the speed of sound in a wire is

Accepted Answer

The speed of sound in a wire or string that is stretched is directly related to the tension in the wire and inversely related to its mass per unit length. Mathematically, this is represented by the formula:
$v = \sqrt{\frac{T}{\mu}}$
where $v$ is the speed of sound in the wire, $T$ is the tension in the wire, and $\mu$ is the mass per unit length of the wire.
When we are told the wire obeys Hooke's law, this means the tension $T$ in the wire is directly proportional to the extension $x$. Hooke's law can be represented as:
$T = kx$
where $T$ is the tension, $k$ is the stiffness (spring constant) of the wire, and $x$ is the extension.
So, the original speed of sound $v$ in the wire can be represented as:
$v = \sqrt{\frac{kx}{\mu}}$
When the extension in the wire is increased to $4x$, the new tension, according to Hooke's Law, becomes $T' = k \cdot 4x = 4kx$. The new speed of sound $v'$ in the wire would thus be:
$v' = \sqrt{\frac{4kx}{\mu}}$
Simplifying this:
$v' = \sqrt{4} \cdot \sqrt{\frac{kx}{\mu}}$
$v' = 2 \cdot \sqrt{\frac{kx}{\mu}}$
Since the original speed $$v = \sqrt{\frac{kx}{\mu}}$$, substituting this into our equation for $v'$ gives us:
$v' = 2 \cdot v$
Thus, the new speed of sound in the wire when the extension is increased to $4x$ is $2v$. Therefore, the correct answer is:
Option B: 2v

Answer

2.5v

Answer

2v

Answer

1.5v

Answer

v

Waves

The extension in a wire obeying Hooke's law is $x. The speed of sound in the stretched wire is v. If the extension in the wire is increased to 4 x$, then the speed of sound in a wire is

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