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## Main Question or Discussion Point

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 [tex]x[/tex] at a time [tex]t[/tex] is given by

[tex]F(x,t)[/tex].

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

I think that the work done by the force in changing the configuration of the string from some [tex]y(x_0,t_0)[/tex] to [tex]y(x,t)[/tex] should be something like

[tex]\int_{y(x_0,t_0)}^{y(x,t)} F(x,t) dy[/tex]

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

[tex]dy=y_t dt+y_x dx[/tex]

where

[tex]\frac{\partial y}{\partial x}=y_x[/tex]

Now the integral becomes something like

[tex]\int F(x,t) y_t dt + \int F(x,t) y_x dx[/tex]

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

Any help would be greatly appreciated :D

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 [tex]x[/tex] at a time [tex]t[/tex] is given by

[tex]F(x,t)[/tex].

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

I think that the work done by the force in changing the configuration of the string from some [tex]y(x_0,t_0)[/tex] to [tex]y(x,t)[/tex] should be something like

[tex]\int_{y(x_0,t_0)}^{y(x,t)} F(x,t) dy[/tex]

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

[tex]dy=y_t dt+y_x dx[/tex]

where

[tex]\frac{\partial y}{\partial x}=y_x[/tex]

Now the integral becomes something like

[tex]\int F(x,t) y_t dt + \int F(x,t) y_x dx[/tex]

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

Any help would be greatly appreciated :D

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