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?

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to determine the lengths of the two strings, L₁ (green string) and L₂ (red string), such that pulses sent simultaneously from the knot reach the ends of the strings at the same time. This requires the pulse travel times to be equal for both strings.
2. Step 2: Recall the formula for wave speed on a string: v = √(T/μ), where v is the wave speed, T is the tension in the string, and μ is the linear density of the string. The linear density for string 1 is given as 2.0 g/m, and the linear density for string 2 is unspecified but can be denoted as μ₂.
3. Step 3: Write the relationship for travel time. The travel time for a pulse is t = L/v, where L is the length of the string and v is the wave speed. For the pulses to reach the ends simultaneously, the travel times for both strings must be equal: t₁ = t₂. Substituting the formula for wave speed, this becomes L₁/√(T/μ₁) = L₂/√(T/μ₂).
4. Step 4: Simplify the equation. Since the tension T is the same for both strings, it cancels out, leaving L₁/√μ₁ = L₂/√μ₂. Rearrange this equation to find the relationship between L₁ and L₂: L₁ = L₂ × √(μ₁/μ₂).
5. Step 5: Use the total length constraint. The total length of the two strings is given as L₁ + L₂ = 3.5 m. Combine this with the relationship from Step 4 to solve for L₁ and L₂. Substitute μ₁ = 2.0 g/m and μ₂ (linear density of string 2) into the equations to find the lengths.

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 lengths required for the pulses to reach the ends simultaneously.

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.