What is the density of the constantan wire?

[Richie Smash]

Homework Statement

The following data concerning a constantan resistance wire is taken from a catalogue: ''Diameter/m = 5.6 x 10^-4, Resistance per meter = 1.947 ohms, Resistance per Kg = 913 ohms''.
Use the data to determine the density of the constantan wire in kg m^-3.

Homework Equations

V= IR
d=m/v
Volume of a cylinder: pi x Rsquared x height

The Attempt at a Solution

Well to be completely honest, I've never seen a question like this and I'm clueless, I really am.
But I still tried, I tried doing 1/0.00056 using the density formula, but my answer seemed way off.
Then I tried finding the radius from the diameter given and substituting it into the volume of a cylinder formula and then once finding the volume, substituting that into the density formula and assuming the mass to be 1 Kg.
But I'm pretty sure it's wrong, I'm getting some pretty wild numbers and I want to solve this :(

 

[haruspex]

1/0.00056 using the density formula

The density formula you quote relates mass, density and volume. How does that become 1/diameter?

tried finding the radius from the diameter given and substituting it into the volume of a cylinder formula and then once finding the volume, substituting that into the density formula

A better start, but to get a volume you must choose a length. Since you are given resistance per metre, you could choose a 1m length of wire.
Please post your working.

 

[Richie Smash]

Ok well, the answer in my book says 8660Kg m ^-3.

My first working was this, 0.00056/2 = 0.00028
Now the Cylinder formula, 0.00028squared x pi x 1 = 0.000000246
SO now I assume that's my volume, and since I'm given resistance per KG, I use 1 Kg as my mass.
So d = 1/0.000000246 = 4065040.65... which doesn't seem correct.

But then I realized resistance per metre, is resistivity, and there is a formula linking resistivity to resistance, which is:

''R = pl/A'' where (p) is resistivity, (l) is the length of the conductor, and (A) is the cross sectional area.

Now I realized I don't have a value for A, so I used the formula, and got this:
913ohms = (1.947 x 1)/A
913(A) = 1.947
A= 1.947/913 = 0.00213253m^2.

So, now having found the area, if I'm using the length as 1, if I substitute it into the volume of a cylinder formula, it would just be the same
0.00213253m^3.

Now I substitute this volume into the density formula d = m/v

I get d = 1/0.00213253 = 468.93 Kg m^-3.

Sigh as you can see I'm still stuck.

 

[BvU]

Hi Richie,

SO now I assume that's my volume

It is the volume of 1 m of wire. You are given it has a resistance of 1.947 $\Omega$. No way you can imagine that it has a mass of 1 kg.

For 1 kg of wire you are given that the resistance of that wire is 913 $\Omega$. So how many metres is that ?

But then I realized resistance per metre, is resistivity, and there is a formula linking resistivity to resistance, which is:

''R = pl/A'' where (p) is resistivity, (l) is the length of the conductor, and (A) is the cross sectional area.

And right after that you should realize that you are not interested in 'p' .

Now I realized I don't have a value for A,

But you do !

Volume of a cylinder: pi x Rsquared x height

Shows that you do know the area of a circle (wires are supposed to be round) !

So you have a number of metres for 1 kg, and can calculate the volume. Then comes d = m/v .

Some advice: work in terms of clearly defined symbols until you have something like "answer = expression". Check the dimensions and only THEN grab a calculator.​

 

[Richie Smash]

Thanks so much dude!

I figured it out by finding how many metres in 1Kg of wire by using the values of resistance I was given, then I used the cylinder formula, and then i finally used the density formula and got 8658.234Kg m^-3. and I know it's not identical to the books answer but that is because I rounded off during my calculations, so I understand why!

 

[BvU]

Your given info is in three digits (3.5 if you look at the 1.947 $\Omega$, so your answer can't be more than three digits.
I get approximately the same result as you do, but 8660 is a better answer than 8658.2323971...

 

[Richie Smash]

Your given info is in three digits (3.5 if you look at the 1.947 $\Omega$, so your answer can't be more than three digits.
I get approximately the same result as you do, but 8660 is a better answer than 8658.2323971...

Oh I see they approximated to 3 S.F.