Thermal Expansion of solids

BunDa4Th

1. The problem statement, all variables and given/known data

A steel ring with a hole having area of 3.96 cm2 is to be placed on an aluminum rod with cross-sectional area of 4.00 cm2. Both rod and ring are initially at a temperature of 36.0°C. At what common temperature can the steel ring be slipped onto one end of the aluminum rod?

2. Relevant equations

DeltaA = yA_0DeltaT
T = T_0 + DeltaT
DeltaT = A_a - A_s/(y_sA_s) - (y_aA_a)

3. The attempt at a solution

first i did 4 - 3.96 = .04 which is DeltaA

knowing that alluminum is 24x10^-6 (*C^-1) and steel is 11x10^-6 (*C^-1)
since the steel ring needs to be expand so it can fit in the alluminum rod i did
.04 = (11x10^-6)(3.96)(DeltaT) solve for DeltaT (i get this real huge number which is incorrect most likely)

T = T_0 + DeltaT when input its incorrect and when i try T = T_0 - DeltaT is also incorrect.

BunDa4Th

would i use this formula to solve for the temperature needed

DeltaA = A_i*alpha*(T2 - T1) solve for T2?

BunDa4Th

Well, I gave this problem another try and still not understanding this, even after reading the book. Any help would really be helpful because this problem would really help me study for the final i have in 2 weeks. BTW, I tried 20 times and all incorrect....

would i have to find the temprature of both

DeltaA = (alpha_s)(A_s)(DeltaT) solve for deltaT
DeltaA = (alpha_a)(A_a)(DeltaT) solve for deltaT

with those two i would do DeltaT_a - DeltaT_s? then T = T_0 - DeltaT?

OlderDan

If alpha represents a linear expansion coefficient then the your DeltaA equations are not correct. You might think about why, but my next comment will give you a big hint. Maybe you should try thinking in terms of the diameter changes needed to make the ring fit.

BunDa4Th

Okay I dont understand why it is incorrect. Wouldn't it need a .04 cm^2 in change to allow the steel ring to fit? Also i got most of those formula and average coefficient of linear expansion from the book i read and from an example i looked at.

OlderDan

What you need is for the areas to become equal. You are heating both things, so they are both going to expand. So no, you are not looking for a .04 cm² change in one of the areas. You are looking for the changes to each area that will make them equal to each other.

The problem with using linear coefficients of expansion to find the change in area is that area is NOT a linear quantity. You need to either come up with the area coefficient of expansion (which you can do by thinking of the expansion of the area of a flat plate in terms of its linear expansions), or you need to compare a linear dimension of the two objects (the diameter or radius will do nicely).

BunDa4Th

okay so what i need to do is heat up the steel ring to get the area of 4 cm^2 to fit at the end of the aluminum rod. Is that correct?

to get deltaA it is DeltaA = A - A_0. To get A it is A = L^2 where

L = L_0 + alpha*L_0DeltaT.

Now if that is the correct way to do it I am much more confuse. as to what L is and I dont understand what do you mean by area coefficient of expansion or copmare linear dimension.

(sorry i took to long to reply, I was working on my design project and trying to find a digital camera battery which only Samsung uses so it was hard to find.)

Doc Al

No. At the initial temperature, the steel ring has a smaller diameter than the aluminum rod. Heat them up, both will expand; Cool them down, both with contract. (Compare their expansion coefficients to determine which expands or contracts more.)

As OlderDan explained, you need to find what final temperature will make them the same size so that one can fit into the other. Find expressions for the size of each as a function of temperature; set those expressions equal and solve for the temperature difference.

Again, as OlderDan explained, since you have linear coefficients of expansion, it may be easier to work with linear dimensions like diameters instead of areas. (Treat the rod and hole as circles--find their initial diameters.)

This may help: Thermal Expansion

BunDa4Th

Okay, I will give this a try again later tonight when i finish up my final project.

Thanks for all the explaining and Link.

BunDa4Th

Well, I gave this problem another try and got lost on where to actually start first then i know i had to set two equation to equal each other so i thought

2alph_aDeltaT = 2alpha_sDeltaT but realize that the deltaT will cancel each other.

so I thought maybe i had to place the initial temperature into the equation but was not sure how since when solving it i will get the same result where deltaT is equal to 1 again.

I am to the point where i just want to give up on this problem and hope it never show up again.....

Doc Al

You need to compare the size, not just the change in size. Your equations should look like:
New Size = Original Size + Change in size

Get two equations like that and set them equal.

BunDa4Th

okay if i did this correctly A =pir^2

steel ring diameter: 2.245
aluminum diameter: 2.257

L_0 +alphaL_0DeltaT = L_0 + alphaL_0DeltaT

Sorry if this is not correct or what you are trying to tell me but this chapter is just really confusing. I am just having a really hard time understanding this chapter.

steel = aluminum

Doc Al

That looks good to me. Now you can solve for Delta T.

BunDa4Th

Okay i solved for deltaT i got -454.80 which is incorrect because that is way to high. Did i do a mistake in the calculation? or is there a step i am missing?

BunDa4Th

Okay, nevermind I figure out what i did wrong and got the answer. I also figure out how to solve it another way.

DeltaT = (A_s - A_a)/((A_a)(2alpha_a) - (A_s)(2alpha_s))

= -.04/(.000192 - .00008712)

DeltaT = -381
T = T_i + DeltaT T= -345

thanks for explaining everything and helping me figure out how to solve this equation.