/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 A \(3.2-\) kg balloon is filled ... [FREE SOLUTION] | 91影视

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Chapter 15: Problem 34

A \(3.2-\) kg balloon is filled with helium (density \(=0.179 \mathrm{kg} / \mathrm{m}^{3}\) ). If the balloon is a sphere with a radius of \(4.9 \mathrm{m},\) what is the maximum weight it can lift?

Short Answer

Expert verified
The balloon can lift a maximum weight of approximately 833.25 N.

Step by step solution

01

Calculate the Volume of the Balloon

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. Substituting the radius of the balloon, \( r = 4.9 \) m, we find: \[ V = \frac{4}{3} \pi (4.9)^3 \approx 492.53 \text{ m}^3 \]
02

Calculate the Weight of the Helium

The weight of the helium can be calculated by multiplying its volume by its density: \[ \text{Weight of Helium} = V \times \text{Density of Helium} = 492.53 \times 0.179 \approx 88.14 \text{ kg} \]
03

Calculate the Buoyant Force

The buoyant force is equal to the weight of the air displaced by the balloon, which is the same as the weight of the helium in this context (assuming similar conditions for air): \[ \text{Buoyant force} = \text{Weight of Helium} \times g = 88.14 \times 9.81 \approx 864.64 \text{ N} \] where \( g \) is the acceleration due to gravity, approximately \( 9.81 \text{ m/s}^2 \).
04

Subtract the Weight of the Balloon

The maximum weight the balloon can lift is the buoyant force minus the weight of the balloon itself: \[ \text{Weight of Balloon} = 3.2 \times 9.81 \approx 31.39 \text{ N} \] \[ \text{Maximum lift} = 864.64 - 31.39 \approx 833.25 \text{ N} \]
05

Final Calculation: Convert the Lift Force to Weight

Since the question asks for the weight that can be lifted, and weight is a force, we can use the value from the previous step, so the maximum weight it can lift is approximately \( 833.25 \text{ N} \).

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Key Concepts

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

Volume of a Sphere
The volume of a sphere is a fundamental concept when dealing with spherical objects like balloons. The volume determines how much space an object occupies. To compute the volume of a sphere, you can use the formula: \[ V = \frac{4}{3} \pi r^3 \]Here, \( V \) represents the volume, \( \pi \) is a constant (approximately 3.14159), and \( r \) is the radius of the sphere.
  • The radius is the distance from the center of the sphere to any point on its surface.
  • In our example, the balloon's radius is 4.9 meters.
By substituting the radius into the formula, you can find that the volume of the balloon is about 492.53 cubic meters (m鲁). Understanding this helps in calculating other properties like density and buoyant force.
Density
Density is a measure of how much mass is contained in a given volume. It is an essential aspect of understanding how substances interact in different environments. The formula for density is:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]In our balloon scenario, density plays a crucial role in determining how much weight the balloon can lift. By knowing the density of helium (0.179 kg/m鲁), we can calculate the weight of the helium inside the balloon.
  • Helium is chosen because it is lighter than air, hence it provides an upward force.
  • This upward force helps lift the balloon and additional weight.
Knowing the density allows us to understand why the balloon can lift weight despite its mass, given that it's filled with a less dense gas than the surrounding air.
Buoyant Force
Buoyant force is a key principle in fluid mechanics, explaining why objects float or sink in a fluid. It is the force exerted by a fluid that opposes the weight of an object immersed in it. According to Archimedes' principle, the buoyant force is equal to the weight of the fluid displaced by the object.
  • In our example, the balloon displaces an equal volume of air.
  • The weight of this displaced air impacts how much additional weight the balloon can lift.
The force can be calculated using the formula:\[ \text{Buoyant Force} = \text{Density of fluid} \times \text{Volume} \times g \]where \( g \) is the acceleration due to gravity (9.81 m/s虏). For our helium-filled balloon, this force allows understanding of the lift capacity.
Acceleration Due to Gravity
Acceleration due to gravity is an important factor in determining forces, particularly weight. On Earth, this constant is approximately 9.81 m/s虏. It represents the rate at which an object accelerates due to the Earth's gravitational pull. In the context of our balloon scenario, gravity plays a critical role in computing the weight-related forces.
  • The gravitational force impacts both the balloon and the objects it can lift.
  • Calculating the weight uses the formula: \( \text{Weight} = \text{Mass} \times g \).
Understanding the acceleration due to gravity helps in ensuring accuracy when determining real-world lifting capacities. This constant ensures that calculations for forces such as weight remain consistent across various scenarios.

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Most popular questions from this chapter

A water storage tower, like the one shown in the accompanying photo, is filled with freshwater to a depth of \(6.4 \mathrm{m}\). What is the pressure at (a) \(4.5 \mathrm{m}\) and (b) \(5.5 \mathrm{m}\) below the surface of the water? \((c)\) Why are the metal bands on such towers more closely spaced near the base of the tower?

Measuring Density with a Hydrometer A hydrometer, a device for measuring fluid density, is constructed as shown in Figure \(15-30 .\) If the hydrometer samples fluid \(1,\) the small float inside the tube is submerged to level \(1 .\) When fluid 2 is sampled, the float is submerged to level 2 . Is the density of fluid 1 greater than, less than, or equal to the density of fluid \(2 ?\) (This is how mechanics test your antifreeze level. since antifreeze [ethylene glycol] is more dense than water, the higher the density of coolant in your radiator the more antifreeze protection you have.)

Predict/Explain A person floats in a boat in a small backyard swimming pool. Inside the boat with the person are some bricks. (a) If the person drops the bricks overboard to the bottom of the pool, does the water level in the pool increase, decrease, or stay the same? (b) Choose the best explanation from among the following: When the bricks sink they displace less water than when they were floating in the boat; hence, the water level decreases. II. The same mass (boat \(+\) bricks \(+\) person) is in the pool in either case, and therefore the water level remains the same. III. The bricks displace more water when they sink to the bottom than they did when they were above the water in the boat; therefore the water level increases.

BIO How Many Capillaries? The aorta has an inside diameter of approximately \(2.1 \mathrm{cm},\) compared to that of a capillary, which is about \(1.0 \times 10^{-5} \mathrm{m}(10 \mu \mathrm{m}) .\) In addition, the average speed of flow is approximately \(1.0 \mathrm{m} / \mathrm{s}\) in the aorta and \(1.0 \mathrm{cm} / \mathrm{s}\) in a capillary. Assuming that all the blood that flows through the aorta also flows through the capillaries, how many capillaries does the circulatory system have?

IP A block of wood floats on water. A layer of oil is now poured on top of the water to a depth that more than covers the block, as shown in Figure \(15-31\). (a) Is the volume of wood submerged in water greater than, less than, or the same as before? (b) If \(90 \%\) of the wood is submerged in water before the oil is added, find the fraction submerged when oil with a density of \(875 \mathrm{kg} / \mathrm{m}^{3}\) covers the block.
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