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.