A string that is under 50.0 N of tension has linear density 5.0 g/m. A sinusoidal wave with amplitude 3.0 cm and wavelength 2.0 m travels along the string. What is the maximum speed of a particle on the string?

Verified step by step guidance

1. Step 1: Convert the linear density from grams per meter to kilograms per meter. Since 1 g = 0.001 kg, the linear density becomes \( \mu = 5.0 \times 10^{-3} \, \text{kg/m} \).
2. Step 2: Calculate the wave speed \( v \) using the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension in the string (50.0 N) and \( \mu \) is the linear density. Substitute the values into the formula.
3. Step 3: Determine the angular frequency \( \omega \) of the wave using the relationship \( \omega = \frac{2\pi v}{\lambda} \), where \( \lambda \) is the wavelength (2.0 m) and \( v \) is the wave speed calculated in Step 2.
4. Step 4: The maximum speed of a particle on the string is given by \( v_{\text{max}} = \omega A \), where \( A \) is the amplitude of the wave (3.0 cm or 0.03 m) and \( \omega \) is the angular frequency calculated in Step 3.
5. Step 5: Substitute the values of \( \omega \) and \( A \) into the formula \( v_{\text{max}} = \omega A \) to express the maximum speed of a particle on the string.

Key Concepts

Tension in a String

Tension refers to the force exerted along the length of a string or rope, which affects how waves propagate through it. In this context, the tension of 50.0 N influences the wave speed and particle motion on the string. Higher tension generally results in faster wave propagation.

Linear Density

Linear density is defined as the mass per unit length of a string, typically expressed in grams per meter (g/m). It plays a crucial role in determining the wave speed on the string, as it affects how much mass is being moved by the tension. In this case, a linear density of 5.0 g/m will influence the maximum speed of particles on the string.

Wave Speed and Particle Motion

The speed of a wave on a string is determined by the tension and linear density, described by the formula v = √(T/μ), where T is tension and μ is linear density. The maximum speed of a particle on the string is related to the wave's amplitude and frequency. Understanding these relationships is essential for calculating the maximum speed of particles in the given wave scenario.