## Tell me more: the same argument with a few equations

Physics Narrative for 11-14

#### The argument repeated, with algebra

We can follow the argument about shifting fluid with the help of a few simple equations:

volume of liquid pushed out of the left hand side = cross-sectional area of left hand side × height dropped

volume of liquid pushed into the right hand side = cross-sectional area of right hand side × height risen

But the volumes must be equal, therefore:

cross-sectional area of left hand side × height dropped = cross-sectional area of right hand side × height risen

Rewrite this as:

area_{left} × height_{left} = area_{right} × height_{right}

You can rearrange this to give:

area_{left}area_{right} = height_{right}height_{left}

So the changes in the heights are just proportional to the cross-sectional areas.