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. Identify the formula that relates wave speed on a string to the tension and mass per unit length of the string. The formula is $v = \sqrt{\frac{T}{\mu}}$, where $v$ is the wave speed, $T$ is the tension, and $\mu$ is the mass per unit length.
2. Recognize that the mass per unit length ($\mu$) of the string remains constant, as the problem does not indicate any change in the string's mass or length.
3. Understand that when the tension $T$ is halved, the new tension $T' = \frac{T}{2}$. Substitute this new tension into the wave speed formula to find the new wave speed $v'$.
4. Substitute $T'$ into the wave speed formula to get $v' = \sqrt{\frac{T'}{\mu}} = \sqrt{\frac{\frac{T}{2}}{\mu}} = \sqrt{\frac{T}{2\mu}}$.
5. Simplify the expression to find the relationship between the new wave speed $v'$ and the original wave speed $v$. Since $v = \sqrt{\frac{T}{\mu}}$, you can express $v'$ in terms of $v$: $v' = \sqrt{\frac{1}{2}} \cdot v = \frac{v}{\sqrt{2}}$.

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 the tension is halved, the force acting on the string decreases, leading to a change in the wave speed. Understanding how tension influences wave propagation is crucial for solving related problems.

Linear Mass Density

Linear mass density (μ) is defined as the mass per unit length of the string. It plays a significant role in determining wave speed, as it affects how much mass the tension must move. In this scenario, since the linear mass density remains constant, any change in wave speed will solely depend on the change in tension.