Exploring the Relationship Between Line Integrals and Mass in Physics

[mrcleanhands]

Wasn't sure which section to put this q in.

Just reading now that f(x,y) can represent the density of a semicircular wire and so if you take a line integral of some curve C and f(x,y) you can find the mass of the wire... makes sense.

What I don't get is that if I then move the wire around the xy axis it's mass will change. What am I missing?

 

[Vorde]

f(x,y) gives you a value of density for all points in the x-y plane. Taking a line integral through that plane is like 'cutting out' a wire from that density plane.

Unless f(x,y) is constant, there is no reason to assume that one wire cut out would be the same density as another.

I think what you're not realizing is that the weight isn't linked to C.

 

[mrcleanhands]

How is weight not linked to C? to find weight/mass you must take a line integral through that plane, and that line integral is C.

What I was confused about is if I take a 3cm wire and position it from (0,0) to (0,3) and then I move it to some other place in the xy plane suddenly it's weight should change right? which is what doesn't make sense to me...

 

[WannabeNewton]

Why would it change? If you move it over then you can simply make a coordinate transformation that brings it back to the original coordinates while leaving the integral unchanged. The coordinate system is simply a computational tool, it won't change the physical amount of mass in the ring.

 

[Vorde]

Because when you take a line integral from (0,0) to (0,3) in a density field z = f(x,y) you're not just finding out what the mass of a 3 cm wire is you're finding out what the mass of a 3 cm wire is when density is defined by z = f(x,y).

In your mind you're thinking that the 3 cm wire already has a predefined mass, and that we use the line integral to find it. Based off of that, you would think that if you moved this wire to another place in the plane that the line integral should evaluate the same because the weight of the wire isn't changing.

But weight is not predetermined without f(x,y) and C. The question is not "what does a 3 cm wire weigh...lets use a line integral to find out", it is "what does a 3 cm wire weigh when every point has density based off of f(x,y)". In the latter case, if the values of f(x,y) change then of course the weight added up with change.