Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Starting from a homework problem:

An aluminum cup of $100 cm^3$ capacity is completely filled with glycerin at $22°C$. How much glycerin, if any, will spill out of the cup if the temperature of both the cup and the glycerin is increased to $28°C$? (The coefficient of volume expansion of glycerin is $5.1x10^4/C°$.)

I find that I have the for efficient of linear expansion for aluminum, but I need to know how the volume of the cup changes. Worse, I don't know the dimensions of the cup.

I think I use the linear expansion equation for metal rod $\Delta L = L \alpha \Delta T$ to find how much taller the cup is after the temperature changed and the volume expansion equation for a solid of liquid $\Delta V = V \beta \Delta T$ but not knowing any of the dimension of the cup I do not see how to determine this?

share|cite|improve this question
Hi, Kurt. Welcome to Physics.SE. Please take note of the FAQ on what questions should not be asked. This question could be rescued by re-writing it in terms of a general principle. But let me suggest an approach. Start with the initial interior volume of the vessel written as $100\text{ cm}^3 = V_0 = \pi r_0^2 h_0$, now write the interior volume of the vessel after it expands in terms of $r_0$, $h_0$, $\alpha$, and $\Delta T$. – dmckee Feb 3 '12 at 2:00
@dmckee: as homework questions go I don't think this one is all that bad. Kurt did lay out his procedure for solving the problem and asked about a specific aspect of it. Sure, perhaps it's not our ideal conceptual question, but I think it's reasonable for a homework question. – David Z Feb 3 '12 at 2:50
Kurt, I took a crack at generalizing the problem. If you don't like my edits you can roll them back or re-edit. – dmckee Feb 3 '12 at 15:13
up vote 2 down vote accepted

To leading order in $\alpha$, the volume expansion of your container does not depend on its shape, and is equal to $\Delta V = 3V\alpha\Delta T$. It is fairly simple to verify this for a cylinder, cube or sphere. Also the expansion coefficient for aluminum is going to be quite a bit smaller than glycerol, so you may be intended to simply neglect the expansion of the container.

share|cite|improve this answer
Actually solving... since the volumes are initially equal $Spill_{glycerin} = \Delta V_{glycerin} - \Delta V_{aluminum} = \left( V_{glycerin} \beta_{glycerin} - V_{aluminum} \beta_{aluminum} \right) \Delta T$ – rudolph9 Feb 3 '12 at 1:23
I've changed the text of the problem a little for generality, and this answer might benefit from an exhibition of how you got the factor of 3 in this case. – dmckee Feb 3 '12 at 15:15

A few things to ask yourself and keep in mind RE Aluminium: - Does the proportionality of the change in Volume matter in this context? - If you know the linear expansion (Remember this is the change in length/area/volume (all scaler units) VS the change in Temp), then you can work out what the final volume is by simply multiplying the initial volume by (expansion co-efficient * dT) - If this has brought you to a point of uncertainty "50/50", do your unit check: Where C - Temperature in Celsius Where d - Delta (Change in Unit) Where V - Volume (V)cm^3 * (Coefficient)1/C * (dT)C. This will leave you with dV in cm^3 Volume capacity change of Aluminium Cup: 100 * (23e-6 * 6) = X Volume change of Glycerin: 100 * (5.1e-4 * 6) = Y Spilt amount (Z) = Y - X You can do the math. All the best, Blake

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.