Refer to Problem 16 and Figure P 14.16 . A hydrometer is to...

Refer to Problem 16 and Figure $\mathrm{P} 14.16 .$ A hydrometer is to be constructed with a cylindrical floating rod. Nine fiduciary marks are to be placed along the rod to indicate densities of $0.98 \mathrm{~g} / \mathrm{cm}^{3}, 1.00 \mathrm{~g} / \mathrm{cm}^{3}, 1.02 \mathrm{~g} / \mathrm{cm}^{3}, 1.04 \mathrm{~g} / \mathrm{cm}^{3}, \ldots$ $1.14 \mathrm{~g} / \mathrm{cm}^{3} .$ The row of marks is to start $0.200 \mathrm{~cm}$ from the top end of the rod and end $1.80 \mathrm{~cm}$ from the top end. (a) What is the required length of the rod? (b) What must be its average density? (c) Should the marks be equally spaced? Explain your answer.

Refer to Problem 16 and Figure $\mathrm{P} 14.16 .$ A hydrometer is to be constructed with a cylindrical floating rod. Nine fiduciary marks are to be placed along the rod to indicate densities of $0.98 \mathrm{~g} / \mathrm{cm}^{3}, 1.00 \mathrm{~g} / \mathrm{cm}^{3}, 1.02 \mathrm{~g} / \mathrm{cm}^{3}, 1.04 \mathrm{~g} / \mathrm{cm}^{3}, \ldots$
$1.14 \mathrm{~g} / \mathrm{cm}^{3} .$ The row of marks is to start $0.200 \mathrm{~cm}$ from the top end of the rod and end $1.80 \mathrm{~cm}$ from the top end.
(a) What is the required length of the rod?
(b) What must be its average density?
(c) Should the marks be equally spaced? Explain your answer.

Key Concepts

Nonlinear Relationship between Immersion Depth and Fluid Density

Even though the density increments marked on the hydrometer are uniform, the corresponding change in immersion depth does not vary linearly with the fluid density. This is a direct consequence of the inverse relationship in the equilibrium condition, which means that the spacing between the fiduciary marks along the hydrometer will not be equal. Recognizing this nonlinearity is important for achieving proper calibration of the device.

Floating Equilibrium of Uniform Objects

For a uniformly dense floating object, such as a cylindrical rod, the equilibrium condition is achieved when the weight of the object equals the buoyant force exerted by the displaced fluid. This involves ensuring that the overall (average) density of the object is less than that of the fluid (when floating) and that the ratio of the immersed length to the total length is correctly related to the densities involved. This concept is essential when determining both the physical dimensions and the average density required for the hydrometer.

Archimedes’ Principle

This principle states that a body fully or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid that it displaces. In the context of a floating hydrometer, it provides the basis for understanding the equilibrium condition where the weight of the hydrometer is balanced by the buoyant force, thus linking the depth of immersion to both the hydrometer’s average density and the density of the liquid.

Hydrometer Design and Calibration

A hydrometer is an instrument used to measure the density of liquids by floating in them and indicating a reading based on how far it is immersed. The calibration of a hydrometer involves marking positions on the instrument corresponding to specific fluid densities. These marks must be carefully placed to accurately reflect the relationship between the immersed length and the density of the liquid, making the proper design critical for accurate measurements.

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