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Chapter 14: Problem 80

In an experiment, a rectangular block with height \(h\) is allowed to float in four separate liquids. In the first liquid, which is water, it floats fully submerged. In liquids \(A, B\), and \(C\), it floats with heights \(h / 2,2 h / 3\), and \(h / 4\) above the liquid surface, respectively. What are the relative densities (the densities relative to that of water) of (a) \(A,(\mathrm{~b}) B\), and \((\mathrm{c}) C?\)

Short Answer

Expert verified
Relative densities: (a) 1/2, (b) 1/3, (c) 3/4.

Step by step solution

01

Understanding Archimedes' Principle

When an object floats, it displaces a volume of liquid equal to its own weight. The buoyant force is equal to the weight of the fluid displaced. The density of the object relative to the fluid determines its submerged depth.
02

Determine Density in Water

Since the block floats fully submerged in water, its density must be equal to the density of water. Let's denote the block's density as \( \rho_{\text{block}} = \rho_{\text{water}} \).
03

Calculate Density relative to Liquid A

In liquid A, the height above the liquid surface is \( h/2 \). This means \( h/2 \) of the block is submerged. Therefore, the fraction submerged is \( 1 - \frac{1}{2} = \frac{1}{2} \). The relative density of liquid A is the same as the fraction submerged: \( \rho_A = \frac{1}{2} \rho_{\text{water}} \).
04

Calculate Density relative to Liquid B

In liquid B, the height above the liquid surface is \( 2h/3 \). Hence, \( h - 2h/3 = h/3 \) is submerged. The fraction submerged is \( \frac{1}{3} \), so: \( \rho_B = \frac{1}{3} \rho_{\text{water}} \).
05

Calculate Density relative to Liquid C

In liquid C, the height above the liquid surface is \( h/4 \). Therefore, the submerged portion is \( h - h/4 = 3h/4 \). The relative density is given by the submerged fraction: \( \rho_C = \frac{3}{4} \rho_{\text{water}} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Buoyant Force
Buoyant force is a fundamental concept in fluid mechanics. It's the upward force exerted by a fluid on any partially or fully submerged object. This force is what makes objects float or immerse in a fluid. According to Archimedes' Principle, the buoyant force acting on an object is equal to the weight of the fluid displaced by that object. This principle explains why objects can float even if they're denser than water, as long as enough fluid is displaced.
  • When an object is immersed in a fluid, it pushes some of the fluid out of the way.
  • The fluid push back, creating an upward force.
  • This force is called the buoyant force, counteracting the weight of the object.
This principle is critical for understanding how ships float and balloons rise. It even plays a role in everyday activities, like swimming.
Density
Density is a key property of materials, indicating how much mass is contained within a given volume. It's commonly expressed in units like grams per cubic centimeter (g/cm鲁) or kilograms per cubic meter (kg/m鲁).
  • The formula to calculate density is \( \rho = \frac{m}{V} \), where \( m \) is mass and \( V \) is volume.
  • An object will float in a fluid if its density is less than that of the fluid.
  • Conversely, if the object's density is greater, it will sink.
Understanding an object's density is crucial in predicting its behavior in fluids. For instance, a dense material will submerge more when placed in a liquid, compared to a less dense one.
Fluid Displacement
Fluid displacement occurs when an object is immersed in a fluid, pushing the fluid out of the way to make space for itself. The volume of the displaced fluid is equivalent to the volume of the submerged part of the object. This is a direct application of Archimedes' Principle, stating the buoyant force on an object is equal to the weight of the fluid it displaces.
  • The displaced fluid's weight is what causes the buoyant force to act.
  • If the volume of the displaced fluid equals the object's weight, it floats.
  • If the displaced fluid volume is less than the object's weight, it sinks.
Fluid displacement is pivotal in designing ships and submarines, ensuring they float successfully or reach intended depths underwater.
Relative Density
Relative density, or specific gravity, is a measure of density compared to a reference substance, usually water for liquids. It gives insight into whether an object would float or sink in water.
  • Relative density is calculated as \( \frac{\text{density of object}}{\text{density of reference fluid}} \).
  • A relative density less than 1 implies the object is less dense than water and will float.
  • A relative density greater than 1 implies the object is more dense and will sink.
Understanding relative density helps compare densities without using units, making it dimensionless. This measure helps in quickly assessing the floating or sinking behavior of an object in comparison to water.

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Most popular questions from this chapter

Water is pumped steadily out of a flooded basement at a speed of \(5.0 \mathrm{~m} / \mathrm{s}\) through a uniform hose of radius \(1.0 \mathrm{~cm}\). The hose passes out through a window \(3.0 \mathrm{~m}\) above the waterline. What is the power of the pump?

Find the pressure increase in the fluid in a syringe when a nurse applies a force of \(42 \mathrm{~N}\) to the syringe's circular piston, which has a radius of \(1.1 \mathrm{~cm}\).

A glass ball of radius \(2.00 \mathrm{~cm}\) sits at the bottom of a container of milk that has a density of \(1.03 \mathrm{~g} / \mathrm{cm}^{3} .\) The normal force on the ball from the container's lower surface has magnitude \(9.48 \times 10^{-2} \mathrm{~N}\). What is the mass of the ball?

(a) If this longnecked, gigantic sauropod had a head height of \(21 \mathrm{~m}\) and a heart height of \(9.0 \mathrm{~m}\), what (hydrostatic) gauge pressure in its blood was required at the heart such that the blood pressure at the brain was 80 torr (just enough to perfuse the brain with blood)? Assume the blood had a density of \(1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). (b) What was the blood pressure (in torr or \(\mathrm{mm} \mathrm{Hg}\) ) at the feet?

Suppose that two tanks, 1 and 2, each with a large opening at the top, contain different liquids. A small hole is made in the side of each tank at the same depth \(h\) below the liquid surface, but the hole in tank 1 has half the cross- sectional area of the hole in tank \(2 .\) (a) What is the ratio \(\rho_{1} / \rho_{2}\) of the densities of the liquids if the mass flow rate is the same for the two holes? (b) What is the ratio \(R_{V 1} / R_{V 2}\) of the volume flow rates from the two tanks? (c) At one instant, the liquid in tank 1 is \(12.0 \mathrm{~cm}\) above the hole. If the tanks are to have equal volume flow rates, what height above the hole must the liquid in tank 2 be just then?
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