The wave speed on a string under tension is 200 m/s. What is the speed if the tension is halved?

Verified step by step guidance

1. Step 1: Recall the formula for the wave speed on a string under tension: $v=\sqrt{\frac{T}{\mu }}$, where $v$ is the wave speed, $T$ is the tension in the string, and $\mu$ is the linear mass density of the string.
2. Step 2: Understand the relationship between wave speed and tension. The wave speed is proportional to the square root of the tension, meaning if the tension changes, the wave speed changes by the square root of the factor by which the tension changes.
3. Step 3: Identify the change in tension. The problem states that the tension is halved, so the new tension is $\frac{1}{2}T$.
4. Step 4: Substitute the new tension into the formula for wave speed. The new wave speed $\mathrm{v\text{'}}$ can be expressed as $\mathrm{v\text{'}}=\sqrt{\frac{\frac{1}{2}}{T}}$. Simplify this expression to find that $\mathrm{v\text{'}}=v\times \sqrt{\frac{1}{2}}$.
5. Step 5: Use the given initial wave speed of 200 m/s to express the new wave speed in terms of the square root factor. The new wave speed is $200\times \sqrt{\frac{1}{2}}$. This is the final expression for the new wave speed.

Key Concepts

Wave Speed on a String

The speed of a wave on a string is determined by the tension in the string and its linear mass density. The relationship is given by the formula v = √(T/μ), where v is the wave speed, T is the tension, and μ is the linear mass density. This means that as tension increases, wave speed increases, and vice versa.

Tension in a String

Tension refers to the force exerted along the length of the string, which affects how quickly waves can travel through it. When tension is increased, the string becomes tighter, allowing waves to propagate faster. Conversely, reducing the tension decreases the wave speed, illustrating the direct relationship between tension and wave propagation.

Linear Mass Density

Linear mass density (μ) is defined as the mass per unit length of the string. It plays a crucial role in determining wave speed, as it affects how much mass the tension must move. While the problem does not change the linear mass density, understanding its role helps clarify why changes in tension directly influence wave speed.