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Chapter 12: Problem 32

A hollow plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of \(0.650 \mathrm{~m}^{3}\) and the tension in the cord is \(1120 \mathrm{~N}\). (a) Calculate the buoyant force exerted by the water on the sphere. (b) What is the mass of the sphere? (c) The cord breaks and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be submerged?

Short Answer

Expert verified
The buoyant force is approximately 6370 N, the mass of the sphere is about 120 kg, and about 97.8% of the sphere's volume will be submerged once the cord breaks and the sphere rises to the surface.

Step by step solution

01

Calculate the Buoyant Force

We know that the buoyant force can be calculated by the formula: \(F_{b} = \rho_{water} * V_{sphere} * g\). Here, \(蟻_{water}\) is the density of water which is \(1000 \,kg/m^{3}\), \(V_{sphere}\) is the volume of the sphere given as \(0.650 \,m^{3}\) and \(g\) is the acceleration due to gravity which is \(9.8 m/s^{2}\). Substituting these values into the formula, we get the buoyant force.
02

Calculate the Mass of the Sphere

The weight of the sphere, \(W\), is given by the sum of the tension in the cord, \(T\), and the buoyant force, \(F_b\). So \(W = T + F_b\). The mass of the sphere, \(m\), is then given by the weight of the sphere divided by the acceleration due to gravity: \(m = W/g\). Using the known values and calculated values, calculate the mass of the sphere.
03

Determine the Fraction of the Sphere's Volume Submerged

We assume that the sphere is in equilibrium when it reaches the surface, meaning its weight is equal to the buoyant force. The volume of water displaced is directly proportional to the submerged volume of the sphere. Hence: \(W = \rho_{water} * V_{submerged} * g\). Here, \(V_{submerged}\) is the volume submerged. Solving for \(V_{submerged}\), we get: \(V_{submerged} = W / (\rho_{water} * g)\). Divide this by the total volume of the sphere to get the fraction submerged.

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Key Concepts

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

Density of Water
Water is a fascinating substance, and its density plays a crucial role whenever objects interact with it. The density of water is typically given as \(1000\, \text{kg/m}^{3}\) at standard conditions. This is useful for various calculations, including buoyancy, as it directly influences the buoyant force.
Density describes the mass per unit volume and is an essential factor in determining whether an object will float or sink.
  • A substance with a lower density than water will generally float, while a higher density means it tends to sink.
  • The density of water can slightly change with temperature, but for most purposes, \(1000\, \text{kg/m}^{3}\) is sufficiently accurate.
In problems involving buoyancy, understanding the density of water helps you calculate the upward force exerted by the liquid on a submerged object.
Volume of Sphere
A sphere is a perfectly symmetrical three-dimensional shape, like a basketball or soap bubble.The volume of a sphere is calculated using the formula: \[ V = \frac{4}{3} \pi r^3 \]where \( r \) is the radius of the sphere. Knowing the sphere鈥檚 volume is crucial perhaps when you want to calculate the buoyant force acting on it if submerged in water.
  • In the context of buoyancy, the greater the volume of the sphere, the larger the buoyant force.
  • For our sphere with a volume of \(0.650\, \text{m}^{3}\), this volume indicates the amount of water it displaces when fully submerged.
The displaced water's weight leads to the rising or sinking of the sphere, depending on other forces and weights acting upon it.
Mass of Sphere
The mass of a sphere must be understood in terms of its weight when discussing its behavior in a fluid.Mass is calculated by dividing the weight of the object by the acceleration due to gravity:\[ m = \frac{W}{g} \]where \( W \) is the weight, and \( g \) is the acceleration due to gravity, typically \(9.8\, \text{m/s}^2\).
This will help determine characteristics such as the tension in the tethering cord or how much of the sphere emerges from the water.
  • Knowing the mass is essential for finding how much the buoyant force offsets the object's weight.
  • In this exercise, the weight must be directly found from the sum of the tension and the buoyant force.
Acceleration due to Gravity
Gravity is a force that acts between any two masses. On Earth, it gives weight to physical objects and causes them to fall back down when thrown upward.
The standard acceleration due to gravity is \(9.8\, \text{m/s}^2\), which means any object falling under gravity alone will increase its velocity by \(9.8\, \text{m/s}\) for every second it falls.
  • This value appears frequently in physics calculations, and it's what allows us to calculate the weight from mass or vice versa.
  • In buoyancy-related problems, the gravitational acceleration helps to equate weight and buoyant forces affecting an object submerged in a fluid.
By understanding gravity鈥檚 influence, you can predict movements and needed counterbalances, like the cord's tension, when objects are submerged in liquid.

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Most popular questions from this chapter

A medical technician is trying to determine what percentage of a patient's artery is blocked by plaque. To do this, she measures the blood pressure just before the region of blockage and finds that it is \(1.20 \times 10^{4} \mathrm{~Pa}\), while in the region of blockage it is \(1.15 \times 10^{4} \mathrm{~Pa}\). Furthermore, she knows that blood flowing through the normal artery just before the point of blockage is traveling at \(30.0 \mathrm{~cm} / \mathrm{s}\), and the specific gravity of this patient's blood is 1.06 . What percentage of the cross-sectional area of the patient's artery is blocked by the plaque?

How does the force the diaphragm experiences due to the difference in pressure between the lungs and abdomen depend on the abdomen's distance below the water surface? The force (a) increases linearly with distance; (b) increases as distance squared; (c) increases as distance cubed; (d) increases exponentially with distance.

If the elephant were to snorkel in salt water, which is more dense than freshwater, would the maximum depth at which it could snorkel be different from that in freshwater? (a) Yes - that depth would increase, because the pressure would be lower at a given depth in salt water than in freshwater; (b) yes - that depth would decrease, because the pressure would be higher at a given depth in salt water than in freshwater; (c) no, because pressure differences within the submerged elephant depend on only the density of air, not the density of the water; (d) no, because the buoyant force on the elephant would be the same in both cases.

A shower head has 20 circular openings, each with radius \(1.0 \mathrm{~mm}\). The shower head is connected to a pipe with radius \(0.80 \mathrm{~cm}\). If the speed of water in the pipe is \(3.0 \mathrm{~m} / \mathrm{s},\) what is its speed as it exits the shower-head openings?

(a) What is the average density of the sun? (b) What is the average density of a neutron star that has the same mass as the sun but a radius of only \(20.0 \mathrm{~km} ?\)
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