Work done by a distributed force on a string

[Eidos]

Hi All

I'd like to know how I could calculate the work done by a distributed force on a string.

Let's say the force at a point x at a time t is given by

F(x,t).

Now the instantaneous amplitude of the string is given by y(x,t), say

I think that the work done by the force in changing the configuration of the string from some y(x_0,t_0) to y(x,t) should be something like
\int_{y(x_0,t_0)}^{y(x,t)} F(x,t) dy

I'll use the total derivative on y(x,t) which gives

dy=y_t dt+y_x dx

where

\frac{\partial y}{\partial x}=y_x

Now the integral becomes something like

\int F(x,t) y_t dt + \int F(x,t) y_x dx

My concerns here are the limits I need to put in each integral.

Any help would be greatly appreciated :D

 

[omoplata]

I would like to ask some questions to understand this better.

How does the force \vec{F}(x,t) act on the string? I'm guessing it's perpendicular to the x direction and along the y direction?

If it is perpendicular, why are you integrating from y(x_{0},t_{0}) to y(x,t) ? Shouldn't it be integrated from y(x,t_{0}) to y(x,t) ? (i.e. instead of \int_{y(x_{0},t_{0})}^{y(x,t)} F(x,t) dy , I think it should be \int_{y(x,t_{0})}^{y(x,t)} F(x,t) dy ). I mean, why does the x coordinate change?

 

[Eidos]

I've attached the picture of how the force enters the system.

I agree with your assertion that the limits should include changes in t only.
How will this effect my change of variable then?

That is when I take the total derivative of dy,
how do I exclude y_x dx from the total derivative?

 

[omoplata]

If the tension in the string is completely neglected, my approach would be

• Find the infinitesimal work dW(x) done between x and x + dx by dW(x) = \int_{y(x,t_{0})}^{y(x,t)} F(x,t) dy. If F doesn't cahnge with time, just take it as F(x). If it does, then find y(t) using Newton's second law for the part of string between x and x+dx with mass m by F(x,t) = m \frac{ \partial^{2} y }{ \partial t^{2} }

• Find the total work by W = \int_{x=x_{1}}^{x_{2}} dW(x) dx

But a string in real life would have elasticity and tension and that analysis would be different. Then there would also be the extra work done to stretch the string.