Solving the Temperature Difference for Steel & Aluminum Rings

[pkossak]

A steel sphere sits on top of an aluminum ring. The steel sphere (a (average
coefficient of linear expansion) = 1.1*10^-5/C) has a diameter of 4 cm at 0
C. The aluminum ring (a = 2.4*10^-5/C) has an inside diameter of 3.9940 cm at 0 C. Closest to which temperature given will the sphere just fall through the ring?

I thought that if I took the a*L*deltaT(aluminum ring) - a*L*deltaT(steel) =
1*10^-4 (negative difference of diameters, sphere - ring), I would get the
right answer. However, I wasn't all that close (ended up being 208 C, and I
got an answer much less than that). I was wondering if there was a better way to approach this problem? Thanks a lot

 

[Doc Al]

You know the diameters of the aluminum and steel at T = 0. Figure out the diameter of each as a function of $\Delta T$. Then solve for the temperature at which the diameters are equal.

 

[pkossak]

This has definitely helped me, but I'm stumped on what to do about the deltaL in thermal expansion equation. And if I do figure that out, what do I do to figure out T?

Would it be deltaL/L*a(steel) + deltaL/L*a(aluminum) = T?

Thank you a ton

 

[Doc Al]

The diameter at temperature T equals the original diameter (at 0 degrees) plus the change in diameter from the temperature increase ($\Delta T = T$):
$$D = D_0 + D_0 \alpha T$$

Use this to solve for the temperature at which both diameters are equal.