Energy in oscillating string

[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.

 

[thomate1]

quez

As a scientist, it is important to understand the concept of energy and how it relates to the motion of an oscillating string. Energy is a fundamental concept in physics and is defined as the ability to do work or cause change. In the case of an oscillating string, energy is required to maintain the motion of the string as it oscillates back and forth.

To solve this problem, we can start by using the formula for the total energy of a simple harmonic oscillator, which is given by:

E = (1/2)kA^2

Where E is the total energy, k is the spring constant, and A is the amplitude of oscillation. In the case of an oscillating string, the tension in the string acts as the restoring force, similar to a spring, and can be represented by the spring constant, k.

We are given the length of the string, the tension, and the frequency of oscillation. From this, we can calculate the spring constant using the formula:

k = (4π^2m)/L^2

Where m is the mass of the string and L is the length of the string. Plugging in the values from the problem, we get:

k = (4π^2*1.27 kg)/(4.24 m)^2 = 2.96 N/m

Now, we can plug in all the values into the energy formula to get the total energy of the oscillating string:

E = (1/2)*(2.96 N/m)*(0.0632 m)^2 = 0.00591 J

This is the amount of energy required to keep the rope oscillating for 1 second. To find the total energy for 6.25 seconds, we simply multiply this value by 6.25:

Etotal = 0.00591 J * 6.25 = 0.0369 J

Therefore, the amount of energy required to keep the rope oscillating for 6.25 seconds is approximately 0.0369 Joules. I hope this explanation helps you understand the relation between energy and oscillating strings.