(II) (a) Show that the average rate with which energy is...

(II) $(a)$ Show that the average rate with which energy is transported along a cord by a mechanical wave of frequency $f$ and amplitude $A$ is $$\overline{P}=2 \pi^{2} \mu v f^{2} A^{2},$$ where $v$ is the speed of the wave and $\mu$ is the mass per unit length of the cord. (b) If the cord is under a tension $F_{T}=135 \mathrm{N}$ and has mass per unit length $0.10 \mathrm{kg} / \mathrm{m},$ what power is required to transmit $120-\mathrm{Hz}$ transverse waves of amplitude 2.0 $\mathrm{cm} ?$

Key Concepts

Frequency and Amplitude Dependence

The energy carried by a wave, and consequently the power transmitted, is strongly dependent on both the amplitude and the frequency of the wave. Specifically, the energy scales with the square of the amplitude and the square of the frequency, meaning that small changes in these parameters can significantly affect the power transmitted along the medium.

Wave Speed on a Tensioned String

The speed at which a mechanical wave travels along a cord or string is determined by the physical properties of the medium, specifically the tension in the string and its mass per unit length. This relationship is given by the formula v = ?(F_T/?), which indicates that higher tension or lower mass density results in a faster propagating wave.

Average Power in a Wave

The average power of a wave is defined as the energy transmitted per unit time. It is calculated by considering the instantaneous power, which is linked to both the kinetic and potential energy of the vibrating medium at any point, and then averaging this value over a complete period of the wave's motion.

Energy Transport in Waves

Mechanical waves transport energy via the oscillatory motion of particles in the medium. The power, or average rate of energy transfer, depends on how much energy is stored in each oscillation (determined by the wave's amplitude) and the speed at which the wave travels.

Mechanical Waves

These are disturbances that propagate through a medium, transporting energy without the bulk movement of the medium itself. In the context of a cord or string, mechanical waves are typically transverse vibrations that move along the string, conveying energy from one point to another.

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