Question

Starting with Equation 1, and applying Equation 2, show that the stretching of rubber to a new length, lx requires a force given by:
F = NkT (λ_x − 1/λ^2_x)(b_0)(t_0)
Equation1:W=1/2NkT(λ^2_x+2/λ_x-3)l_0 b_0 t_0
Equation 2: F=dW/dl_x

Answer 1

the notation is a bit confusing. I'm pretty sure I have the right equations, but do you think you could try typing them with the equation editor?

Answer 2

\[F=NkT(\lambda _{x}-\frac{ 1 }{ \lambda ^{2}_{x} })b _{0}t _{0}\]

Answer 3

that's useful, i had it wrong haha. if you could do that i'll take a look.

Answer 4

\[W=\frac{ 1 }{ 2 }NkT(\lambda _{x}^{2}+\frac{ 2 }{ \lambda _{x} }-3)l _{0}b _{0}t _{0}\]

Answer 5

Equation 2: \[F=\frac{ dW }{ dl _{x} }\]

Answer 6

would you be able to help me?

Answer 7

I think the main idea is to take the derivative of W with respect to l_x. to do that, you need to rewrite the mass density lambda as m/l_x

Answer 8

you mean like this \[w=\frac{ 1 }{ 2}NkT(\frac{ m }{ lx }+\frac{ 2 }{ \frac{ m }{ lx } }-3)l _{0}b _{0}t _{0}\]

Answer 9

yep. don't forget the square on the first term

Answer 10

oh yea...so is that the answer?

Answer 11

No, you have to take the derivative of the W with respect to l_x now

Answer 12

how would i do that?

Answer 13

Everything is a constant. You simply derivate any l_x you see.

Answer 14

i dont get it

Answer 15

I actually just did the derivation and it's wrong. I don't think lamda is mass density. It's something called extension ratio. It's l_x / l_0

Answer 16

I'll try to walk you through that with the new info

Answer 17

So we can write the work equation as:\[W=\frac{1}{2}NkT(\frac{l_{x}^{2}}{l_{0}^{2}}+\frac{2l_{0}}{l_{x}}-3)l_{0}b_{0}t_{0}\]

Answer 18

ok

Answer 19

no we derive?

Answer 20

Yep, with respect to l_x

Answer 21

do we use like chain rule?

Answer 22

Nope, you just need to do each term separately. Only a product would need chain rule.

Answer 23

i dont really know how to do it

Answer 24

Okay I'll go through it, just give me a moment.

Answer 25

I keep going through it and i'm ending up with an extra factor that shouldn't be there. So i'm reserving typing this out until I get the right thing.

Answer 26

I keep getting:\[W=\frac{NkT}{l_{0}^{2}}(\lambda _{x}-\frac{1}{\lambda_{x}^{2}})b_{0}t_{0}\]
I don't know where I'm getting the extra factor of l_o^2

Answer 27

is this the answer?

Answer 28

It shouldn't have the lo^2 in the denominator. It should be what it says you should get for F. i wrote W by accident. Should be an F.