A 0.40-kg cord is stretched between two supports 8.7 m apart. When one support is struck by a hammer, a transverse wave travels down the cord and reaches the other support in 0.85 s. What is the tension in the cord?

Verified step by step guidance

1. Determine the speed of the transverse wave using the formula for wave speed: $v=\frac{d}{t}$, where $d$ is the distance the wave travels (8.7 m) and $t$ is the time it takes (0.85 s).
2. Use the relationship between wave speed, tension, and linear mass density: $v=\sqrt{\frac{T}{\mu }}$, where $v$ is the wave speed, $T$ is the tension in the cord, and $\mu$ is the linear mass density.
3. Calculate the linear mass density $\mu$ using the formula: $\mu =\frac{m}{L}$, where $m$ is the mass of the cord (0.40 kg) and $L$ is its length (8.7 m).
4. Rearrange the wave speed formula to solve for tension: $T=\mu v^2$. Substitute the values of $\mu$ and $v$ into this equation.
5. Simplify the expression to find the tension $T$ in the cord. Ensure all units are consistent (e.g., meters, kilograms, seconds) throughout the calculation.

Key Concepts

Wave Speed

Wave speed is the distance a wave travels per unit of time. It can be calculated using the formula v = d/t, where v is the wave speed, d is the distance traveled, and t is the time taken. In this scenario, the wave travels 8.7 m in 0.85 s, allowing us to determine the speed of the wave in the cord.

Tension in a Cord

Tension is the force exerted along the length of a cord or string when it is pulled tight. The tension in a cord affects the speed of a wave traveling through it, as described by the formula v = √(T/μ), where T is the tension and μ is the linear mass density of the cord. Understanding how tension influences wave speed is crucial for solving the problem.

Linear Mass Density

Linear mass density (μ) is defined as the mass per unit length of a cord, calculated as μ = m/L, where m is the mass and L is the length. In this case, the cord has a mass of 0.40 kg and a length of 8.7 m. Knowing the linear mass density is essential for calculating the tension in the cord using the wave speed.