A hydrometer is an instrument used to determine liquid...

A hydrometer is an instrument used to determine liquid density. A simple one is sketched in Figure $\mathrm{P} 14.16 .$ The bulb of a syringe is squeezed and released to let the atmosphere lift a sample of the liquid of interest into a tube containing a calibrated rod of known density. The rod, of length $L$ and average density $\rho_{0}$, floats partially immersed in the liquid of density $\rho$. A length $h$ of the rod protrudes above the surface of the liquid. Show that the density of the liquid is given by $\rho=\frac{\rho_{0} L}{L-h}$

Key Concepts

Density

Density, defined as mass per unit volume, is a key parameter in determining how much of a floating object is immersed in a fluid. In the context of the hydrometer, the known density and dimensions of the rod are used to relate the total weight of the rod to the weight of the displaced fluid, forming the basis for calculating the unknown fluid density.

Fluid Displacement

The concept of fluid displacement involves the volume of fluid that is 'pushed aside' or displaced by an immersed part of an object. The amount of displaced fluid, which directly relates to the submerged portion of the object, is crucial in applying Archimedes' Principle to determine the fluid’s density by balancing forces.

Archimedes' Principle

This principle states that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the body. In problems involving floating objects like a hydrometer, it is used to set the weight of the displaced liquid (which depends on the submerged volume and the fluid's density) equal to the weight of the object, allowing one to relate immersed volume to densities.

Buoyancy Equilibrium

For an object floating in a fluid, buoyancy equilibrium is the condition where the upward buoyant force exactly balances the downward gravitational force (weight) of the object. This equilibrium gives rise to relationships between the submerged fraction of the object's volume and its density relative to the fluid’s density.

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