Thank you for using the site. I'll be happy to help with this problem.
I'll have the solution (with work shown) posted for you just as soon as I finish typing it up.
Can you please check the numbers that were given in the problem statement?
With the values that are given, the change in volume due to cooling exceeds the original volume of the sphere, which doesn't make any sense.
Yes, the coefficient of thermal expansion is usually assigned the variable name "alpha", which is a Greek letter "a".
But it's the values rather than the variable names that are a problem here.
Does this equation look familiar:
∆V = 3V(alpha)∆T
That is the usual formula for the change in volume. Perhaps your course is teaching something different.
The most suspicious number is ***** value of alpha. Most coefficients of thermal expansion are in the range of 1 x 10^-6 to 200 x 10^-6, so a value of 3.071 x 10^-2 is HUGE.
That equation is little different from linear thermal expansion. It only has the "3" added to it because it is calculating volume instead of length.
Is there an example problem from your textbook or from class that would help me know what approach your instructor is expecting you to use?
That's the difficulty that I am having.
The initial volume is:
V(initial) = (4/3)*pi*(80 cm)^3 = 2,144,660.58 cm^3
According to the formula (just dealing with the thermal compression for now):
∆V = V(final) - V(initial) = 3 * V(initial) * alpha * (Tf - Ti)
V(final) = V(initial) + 3 * V(initial) * alpha * (Tf - Ti)
After I plug in all the values, I get:
V(final) = 2,144,660.58 + 3(2,144,660.58)(3.071 x 10^-2)(-195.79 - 28.65)
V(final) = -42,201,895.70 cm^3
But the final volume can't be negative, so something isn't right.
I have no idea what your teacher may be trying to do. But I have seen a variety of teachers teach unusual things, so there is always the possibility that I'm just not familiar with the particular approach that you are being taught.
What textbook are you using? Perhaps I can find it online.
Ok. Perhaps it's just a typo somewhere, or perhaps he didn't work the problem out in advance. These things happen.
If you get any new information, please feel free to let me know. I'll be happy to revisit the problem.
The difficulty is that the three answers depend on each other. You need the answer to the volume question to calculate the new density for part 2, and then you need that new density to calculate the speed of sound in part 3.
It would be one thing if we were coming up with a volume that was just unusual, like a 95% change in volume. But the result that we are getting is an impossible result. The volume cannot be negative, so the minimum volume would have to be 0, which would make the density infinite (since you can't divide by 0), and then the speed of sound would be 0, since you would be dividing the bulk modulus by an infinite density.
In essence, this process would be creating a black hole. Perhaps that is what your instructor is leading you toward, even though that seems a bit silly. (Does he seem like the sort that would give you a "trick" question like this?)
I can write up a solution that way if you'd like. Alternatively, if you can send me some of the equations from the section of your textbook where this kind of topic is covered, perhaps I can figure out what they are thinking.