Refer to Problem 26 and Figure P15.26. A hydrometer is to be...

Refer to Problem 26 and Figure P15.26. 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 26 and Figure P15.26. 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

Calibration of Measuring Instruments

Calibration is the process of marking or adjusting an instrument so that its readings correspond correctly to physical quantities. For a hydrometer, this means placing fiduciary marks at positions that accurately represent specific fluid densities, ensuring that the device yields reliable measurements.

Average Density Calculation

The concept of average density involves determining the mass-to-volume ratio of an object, which is critical when designing a hydrometer. The average density of the hydrometer must be carefully chosen relative to the fluid for it to float at the desired levels, allowing for precise density readings.

Scaling and Mark Spacing

Scaling involves the geometric and numerical design of the measurement scale on a device. In the case of a hydrometer, the spacing of the fiduciary marks must account for the non-linear relationship between immersion depth and displaced fluid volume. Understanding whether the marks should be equally spaced or not is key to ensuring that each mark consistently represents a specific change in fluid density.

Archimedes' Principle

This principle states that a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid it displaces. It is fundamental to understanding how a floating object, such as a hydrometer, reaches equilibrium, and it provides the basis for relating the object’s immersion depth to its density relative to the fluid.

Hydrostatic Equilibrium

Hydrostatic equilibrium refers to the state in which the buoyant force on a floating object balances its weight. In the context of a hydrometer, proper equilibrium allows the device to float in such a way that specific marks along its body correspond to known fluid densities, ensuring accurate density measurement.

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