How Much Energy Keeps a Rope Oscillating for 6.25 Seconds?

[danvazb]

Homework Statement

A 4.24 m long, 1.27 kg rope under a tension of 475 N oscillates with a frequency of 11.2 Hz. If the oscillation amplitude is 6.32 cm, how much energy is required to keep the rope oscillating for 6.25\,{\rm s?}

Homework Equations

The Attempt at a Solution

I don't quite understand the relation between energy and the traveling wave. Where should I start to solve a problem such as this? I would like to actually do it myself, but could use some initial explanation.
Thank you,
Dan Vaz

 

[AEM]

Perhaps this will help. The power, or rate of flow of energy, in a string is not constant. This is because the power input oscillates. As the energy is passed along the string, it is stored in each piece of the string as a combination of kinetic and potential energy due to the deformation of the string. The power input to the string is often taken to be the average over one period of oscillation and is computed by

$$<P> = \frac{1}{\tau} \int_t ^{t_+\tau} P dt$$

where $$\tau$$ is the period of the oscillations and < > indicates the average. Using the fact that the average value of $$sin^2x$$ or $$cos^2x$$ is 1/2 the average rate of energy flow along a string can be calculated to be

$$<P> = 2 \pi^2 A^2 \nu^2 \frac{T}{v}$$

Since

$$v = \sqrt{ \frac{T}{\mu}}$$

where T is the tension and $$\mu$$ is the mass per unit length, the power can be written as

$$<P> = 2 \pi^2 A^2 \nu^2 \mu v$$

I didn't define it, but $$\nu$$ above is the frequency.