In the laboratory, you are given a cylindrical beaker containing a fluid and you are asked to determine the density $\rho$ of the fluid. You are to use a spring of negligible mass and unknown spring constant $k$ that is attached to a vertical stand.

Question

In the laboratory, you are given a cylindrical beaker containing a fluid and you are asked to determine the density $\rho$ of the fluid. You are to use a spring of negligible mass and unknown spring constant $k$ that is attached to a vertical stand.(a) An irregularly shaped object of known mass $m$ and density $D$ $(D \gg \rho)$ hangs from the spring, but is NOT yet submerged in the liquid. Explain how you could experimentally determine the spring constant $k$. You may also choose from among the following items to complete this task: a metric ruler, a stopwatch, string.
, (b) The spring-object system is now arranged so that the object (but none of the spring) is immersed in the unknown fluid as shown in the diagram above. Describe any changes that are observed in the spring-object system and explain why they occur.
, (c) Explain how you could experimentally determine the density of the fluid.
, (d) Show explicitly, using equations, how you will use your measurements to calculate the fluid density $\rho$.
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Jason · Accepted Answer

Check the explanation for the complete solution. The following is a condensed version of the solution:

Measure the length of the spring without the object and then with the object at rest in air. Their difference is the displacement $\Delta x_{\text{air}}$. Solve for $k$ by algebraically rearranging the force equilibrium equation.
The observed change is a decrease in the spring extension because the fluid’s buoyant force partially offsets the weight of the object.
Using the metric ruler, record the spring displacement when the object is in air and when it is immersed in the fluid. According to Archimedes’ principle, the buoyant force equals the weight of the displaced fluid.
$\boxed{\rho = D \left(1 &#8211; \frac{\Delta x_{\text{fluid}}}{\Delta x_{\text{air}}}\right)}$

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