91Ó°ÊÓ

Sinusoidal waves, like the one traveling along a tensioned string in our exercise, are characterized by their amplitude, wavelength, and frequency. The amplitude of a sinusoidal wave refers to the maximum displacement of a point on the medium from its resting position. It is often denoted as 'A' and measured in meters. A higher amplitude means that the medium's particles move further from their equilibrium position during each wave cycle.

In practical terms, the amplitude is indicative of the energy carried by the wave. The greater the amplitude, the more work is done by the medium's particles as they oscillate, which implies more energy being transferred. In our case, the amplitude is given as 3.0 cm, which is crucial for understanding the maximum speed a particle on the string can achieve as the wave propagates, as it affects the wave's energy and thus the particle's kinetic energy.

Linear density is the measure of mass per unit length, which is critically important in calculations involving waves on strings or cables. It is denoted by the Greek letter 'μ' and is usually given in mass per length (for example, g/m or kg/m). Before it can be used in most wave equations, linear density must be in the SI unit of kg/m.

Our problem involves a conversion from grams per meter to kilograms per meter, since 1 kilogram is equal to 1,000 grams. For precise calculations, we use the conversion: \( \text{Linear Density} (kg/m) = \text{Linear Density} (g/m) \times 10^{-3} \). In the context of our exercise, converting the given linear density of 5.0 g/m to kilograms per meter results in 5.0 x 10^{-3} kg/m, enabling us to calculate the speed of the wave along the string accurately.

To calculate the maximum speed of a particle on a string due to a traveling wave, we need to consider the wave's frequency and amplitude. The maximum speed, denoted Vmax, is the highest velocity that a particle on the medium will achieve as the wave passes through its equilibrium position.

The formula for calculating this maximum speed is given by \( V_{\text{max}} = 2 \times \text{Ï€} \times f \times A \), where 'f' is the frequency of the wave and 'A' is its amplitude. This formula derives from the harmonic motion of the particles in the medium, where the speed is at its maximum when the particles pass through equilibrium position. By substituting the frequency and amplitude into this equation, we obtain the value of maximum speed.

Following this approach in our problem, where the frequency is calculated as 5Hz, and the amplitude is converted to 0.03m, the resulting maximum speed of a particle on the string is approximately 9.42 m/s, showing how both the frequency and amplitude directly influence a wave's effect on particle motion.