What is the initial length of rods A and B in rod C?

[william313]

Hello. I have this easy question in linear expansion. Yeah, it's easy but I am really dumb in physics. Please help..

Homework Statement

Rod A which is 30 cm long, expands by 0.045 cm when heated from 0 degrees Celsius to 100 degrees Celsius. Rob B also 30 cm long expands by 0.075 cm for the same change in temperature. Rod C, also 30 cm long is made up of the materials of Rod A and B, connected end to end. It expands by 0.065 cm when heated from 0 degrees Celsius to 100 degrees Celsius. Calculate the initial lengths of rods A and B in C.

given:
Initial length of A = 30 cm
Change in length of A = 0.045 cm
Initial length of B = 30 cm
Change in length of B = 0.075 cm
Initial length of C = 30 com
Change in length of A = 0.065 cm
Change in temperature = 100 degrees Celsius (100 - 0)
Initial length of Rod A in Rod C = ?
Initial length of Rod B in Rod C = ?

Homework Equations

Coefficient of linear expansion = change in length / (initial length x change in temperature)

The Attempt at a Solution

My strategy is:
- get the coefficient of linear expansion of rods A and C.
- get the percentage of the coefficient of linear expansion of Rod A from Rod C.
- get the value of the change in length of A base on the percentage.
- solve for initial length of Rod A base on the formula given above.
- deduct the initial length of Rod C from the initial length of Rod A to get Rod B's initial length.

solution:

let Li be the initial length. Lia for initial length of rod a, Lib for rod b and Lic for rod c.
let cL be the change in length. cLa for rod a, clb for rod b and cLc for rod c.
let C be the coefficient of linear expansion. Ca for rod a and Cc for rod c
let t be the change in temperature.

C = cL / Li x t
Ca = cLa / Lia x t
Ca = 0.045 cm / 30 cm (100)
Ca = 1.5 x 10^-5

Cc = cLc / Lic x t
Cc = 0.065 cm / 30 cm (100)
Cc = 2.167 x 10^-5

get the percentage of rod a from rod c base on the coefficient of linear expansion:
= 1.5 x 10^-5 / 2.167 x 10^-5
= 0.69

cLa in rod c = 0.065 (.69)
cLa in rod c = 0.045

Lia in rod c = cLa in rod c / Ca x t
Lia in rod c = 30 cm

Lib in rod c = 30 cm - 30 cm
Lib in rod c = 0


As you can see. my attempted solution is obviously an error. Please help. Do you guys know how to answer this?

 

[william313]

bump..
is the length of the post scaring people??
:( :(

 

[vela]

It's not obvious to me that the relative lengths of the pieces are in the same proportion to the coefficients of expansion. Try letting A be the length of material A in rod C and B be the length of material B in rod C. One equation that relates them is that A+B=30 cm. Write another equation based on the expansion of rod C. You should end up with two equations and two unknowns that you can solve.

 

[william313]

^^ thanks.
I haven't done 2 equations with two unknowns before. Can you show me how to do that?

 

[Borek]

http://en.wikipedia.org/wiki/Simultaneous_equations#Substitution_method

 

[vela]

I'm sure you have. You learned how to solve them in first-year algebra.