We\'re supposed to find the mass and volume of an aluminum block from informatio

Question

We're supposed to find the mass and volume of an aluminum block from information found in a lab.

Here's the procedure: We had an aluminum block (no mass given) and a spring of length L. We found the length of the spring (L1) when the aluminum block was on it, and the length of the spring (L2) when the spring and block were in a container filled with water.

So, given L, L1, L2 and the density of aluminum (given as 2700 kg/m^3), how can we find the mass and volume of the block? Is it possible without more information?

Explanation / Answer

The answer is, "it depends." First of all, we need to assume a massless spring - you put the spring in the water. The springs own weight will be less by the buoyant force of the water, which will make the spring a little shorter when it is in the water, unless we disregard this (small) length difference. Next time only put the aluminum block in the water. If you have an extension spring with an initial tension, then you need more information. In this case, the spring would be totally compressed to begin with and it would still be the same length when a small weight was hung from it. You need to overcome the initial tension to make the spring start to extend. So, if the spring looked like a cylinder to begin with, you need more information. On the other hand, if you could see daylight between the coils, you can measure mass and volume. In this case every weight added to the spring makes the spring longer. So, if we can assume a massless spring and if it starts out with open coils F = -kx (L1 - L)k = g*density*volume ---(EQN1) (L2 - L)k = g (density - density water)*volume ---(EQN2) We know L, L1, L2, g, density, water density (1000kg/m^3). We don't know volume and spring constant. So we have two equations and two unknowns (L1 + L2 - L - L)k = g*volume*(2*density - density water) (L1 + L2 - 2*L)*k = 9.8*4400*v k = 9.8*4400*v/(L1+L2-2*L) and then substitute this into EQN1 (L1-L)*k = g*density*volume volume = (L1-L)*k/(g*density) as a check of our algebra, we would also substitute into EQN2 to make sure we have the same answer.

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