Energy of pulse on a string
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Here is a string problem I'm having much trouble with. An explanation would be greatly appreciated.
It's written much more clearly on the attachment, so please go there first!!!
Here it is,
Consider a string under a tension T, mass density ρ with a pulse
ξ(x,t) = ξ_0^ (-(x-ct)^2/a^2)
.
propagating with a speed c. Calculate the kinetic and potential energy in the pulse. Assume the string is real long, I mean really, really, really long, like it stretches from to
(Hint: If you are doing the problem correctly you will need the integral from negative to positive infiniti of e^-x = pi^0.5 at some point in your calculation.
The parameters are any givens m, g, L, etc...
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Solution Summary
The energy of pulse on a string are discussed. The kinetic and potential energy in the pulse is calculated.
Solution Preview
We are dealing with a transversal wave of the form:
ξ(x,t) = ξ_0 Exp[-(x-ct)^2/a^2]
Here ξ is the local perturbation in the height of the string. Now, let's first review some of the physics involved here. The tension is defined as follows. If you consider an arbitrary point on the string, then the part to the left of the string pulls with some force on the part to the right of the string. This force is called the tension. Of course, by action = - reaction the part to the right pulls with equal but opposite force on the part to the left.
Now consider a length element of dx of the string. The total force acting on this length element comes from the tension on both sides of the length element. Now, if this length element were completely straight then the forces acting on both sides would cancel out. Let's focus on the vertical component of the force. The part of the string on the left of the length element exerts a force of T sin(alpha) in the downward direction. For small ξ this is thus T dξ/dx. The part on the right exerts an upward force of dξ/dx. Now, in general, dξ/dx won't be constant, so the two ...
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