Problem 46 - University Physics with Modern Physics

Question

A very large balloon with mass \(M=10.0 \mathrm{~kg}\) is inflated to a volume of \(20.0 \mathrm{~m}^{3}\) using a gas of density \(\rho_{\text {eas }}=\) \(0.20 \mathrm{~kg} / \mathrm{m}^{3}\). What is the maximum mass \(m\) that can be tied to the balloon using a \(2.00 \mathrm{~kg}\) piece of rope without the balloon falling to the ground? (Assume that the density of air is \(1.30 \mathrm{~kg} / \mathrm{m}^{3}\) and that the volume of the gas is equal to the volume of the inflated balloon).

Step-by-Step Solution

Answer

The density of air is 1.30 kg/m³, and the volume of the balloon is 20.0 m³. Answer: To find the maximum mass that can be tied to the balloon, first compute the weight of the air displaced by the balloon using the formula: Weight of air = Density of air × Volume of air × acceleration due to gravity (g). Next, calculate the total weight of the balloon, rope, and lifted mass using the formula: Total weight = (Mass of balloon + Mass of rope + Mass of lifted mass) × g. Finally, equate the buoyant force to the total weight and solve for m. The maximum mass that can be tied to the balloon using a 2.00 kg piece of rope is approximately 25.0 kg.

Step 1: Compute the weight of the air displaced by the balloon

To find the buoyant force, first, calculate the weight of the air displaced by the balloon. The weight of the air can be calculated using the following formula: Weight of air = Density of air × Volume of air × acceleration due to gravity (g) Here, the density of air is given as 1.30 kg/m³, and the volume of the balloon is 20.0 m³. The acceleration due to gravity, g, can be taken as approximately 9.81 m/s². Weight of air = 1.30 kg/m³ × 20.0 m³ × 9.81 m/s² #Step 2: Calculate the total weight of the balloon, rope, and lifted mass#

Step 2: Compute the total weight of the balloon, rope and lifted mass

To determine the maximum mass that can be tied to the balloon without it falling to the ground, we need to find the combined weight of the balloon, the rope, and this maximum mass. The total weight can be calculated with the following formula: Total weight = (Mass of balloon + Mass of rope + Mass of lifted mass) × g Given the mass of the balloon (M) is 10.0 kg and the mass of the rope is 2.00 kg. Let m be the maximum mass that can be lifted by the balloon. Then, we can write the total weight as: Total weight = (10.0 kg + 2.00 kg + m) × 9.81 m/s² #Step 3: Equate the buoyant force to the total weight and solve for m#

Step 3: Equating the buoyant force to the total weight and solving for m

The buoyant force must be equal to the total weight for the balloon to remain afloat. Thus, we can write the following equation: Weight of air = Total weight Substitute the values from steps 1 and 2 into the equation and solve for m: 1.30 kg/m³ × 20.0 m³ × 9.81 m/s² = (10.0 kg + 2.00 kg + m) × 9.81 m/s² Solve the equation for m to find the maximum mass that can be tied to the balloon without it falling to the ground. Remember to include the mass of the rope in the final answer, as the problem asks for the maximum mass that can be tied to the balloon using a 2.00 kg piece of rope.

Key Concepts

Density of Gases

Density is a measure of how much mass is contained in a given volume. For gases, density can vary significantly depending on factors like temperature and pressure. In the context of the balloon problem, we look at the density of the gas inside the balloon, which is given as \( \rho_{\text{gas}} = 0.20 \, \text{kg/m}^3 \). This tells us that for every cubic meter of the gas, it weighs 0.20 kilograms.
On the other hand, the density of the air outside is higher, \( \rho_{\text{air}} = 1.30 \, \text{kg/m}^3 \). This larger density of air compared to the gas inside the balloon is crucial for producing a buoyant force. When the gas inside the balloon is less dense than the air surrounding it, the balloon experiences an upward force that makes it float.

• The difference in density between the gas inside and the air outside determines the buoyant force.
• Lower density gases are often used in balloons because they are lighter than air, helping them rise.

Volume Displacement

Volume displacement refers to the concept of how much space an object takes up in a fluid, like air. When the balloon is inflated, it displaces a volume of air equivalent to its own volume. In this problem, that volume is 20 m³.
This displacement is directly related to the buoyant force. According to Archimedes' principle, the buoyant force on a submerged object is equal to the weight of the fluid displaced by the object. Here, the fluid is air, and the weight of the air displaced is what provides the lift or buoyancy to the balloon.

• The buoyant force arises because the balloon displaces a heavier fluid (air) with a lighter one (gas inside the balloon).
• The greater the volume displaced, the greater the buoyant force.

Acceleration due to Gravity

Acceleration due to gravity \( g \) is the rate at which objects accelerate when falling freely under the influence of Earth's gravitational pull. Its standard value is approximately 9.81 m/s². In calculations of buoyant forces and weights, \( g \) is used to convert mass into weight.
Consider this exercise: we calculate the weight of the air displaced by using \( g \). Similarly, \( g \) is used to determine the weight of the balloon, the rope, and any additional mass added. By ensuring that the total weight is balanced with the buoyant force (produced by the air displaced), the system remains in equilibrium.

• Weight = mass \( \times g \), showing the importance of \( g \) in converting mass measurements to real-world forces.
• Consistent use of \( g \) across calculations ensures accurate representations of forces in physics problems related to buoyancy.