String 1 in FIGURE P16.47 has linear density 2.0 g/m and string 2 has linear density . A student sends pulses in both directions by quickly pulling up on the knot, then releasing it. What should the string lengths L₁ and L₂ be if the pulses are to reach the ends of the strings simultaneously?

Key Concepts

Wave Speed in Strings

The speed of a wave traveling through a string is determined by the tension in the string and its linear density. The formula for wave speed (v) is given by v = √(T/μ), where T is the tension and μ is the linear density. This relationship is crucial for understanding how different strings will transmit wave pulses at different speeds based on their material properties.

Linear Density

Linear density (μ) is defined as the mass per unit length of a string, typically expressed in grams per meter (g/m). It affects the wave speed; a higher linear density results in a slower wave speed for a given tension. In this problem, the linear densities of the two strings are essential for calculating the required lengths to ensure simultaneous arrival of wave pulses.

Simultaneous Wave Propagation

For two wave pulses to reach the ends of their respective strings simultaneously, the time taken for each pulse to travel its length must be equal. This can be expressed mathematically as L₁/v₁ = L₂/v₂, where L is the length of the string and v is the wave speed. By manipulating this equation, one can determine the necessary lengths of the strings based on their linear densities.