Problem 62 - University Physics with Modern Physics

Question

A hot-air balloon has a volume of 2200 \(\mathrm{m}^{3} .\) The balloon fabric (the envelope) weighs 900 \(\mathrm{N} .\) The basket with gear and full propane tanks weighs 1700 \(\mathrm{N} .\) If the balloon can barely lift an additional 3200 \(\mathrm{N}\) of passengers, breakfast, and champagne when the outside air density is \(1.23 \mathrm{kg} / \mathrm{m}^{3},\) what is the average density of the heated gases in the envelope?

Step-by-Step Solution

Answer

The average density of the heated gases is approximately \(0.9621 \, \text{kg/m}^3\).

Step 1: Calculate the Total Lift Required

To determine the average density of the heated gases, we need to first understand the total lift that the air inside the balloon must generate. This lift must support the combined weight of the envelope, basket with gear, passengers and additional items. The total lift required is the sum of these weights: \(900 \, \text{N} + 1700 \, \text{N} + 3200 \, \text{N} = 5800 \, \text{N}\).

Step 2: Calculate the Buoyant Force

The buoyant force exerted by the air on the balloon is equal to the weight of the air displaced by the balloon. We can calculate this using the formula \( F_b = \rho_{\text{air}} \times g \times V\), where \( \rho_{\text{air}} = 1.23 \, \text{kg/m}^3 \) is the density of the outside air, \( g = 9.81 \, \text{m/s}^2 \) is the acceleration due to gravity, and \( V = 2200 \, \text{m}^3 \) is the volume of the balloon. Thus, \( F_b = 1.23 \times 9.81 \times 2200 = 26562.06 \, \text{N}\).

Step 3: Calculate the Weight of the Air in the Balloon

Since the buoyant force is known and the weight it must lift is known, we can find the weight of the air inside the balloon. The weight of the air inside the balloon must be equal to the buoyant force minus the total lift required: \( W_{\text{air}} = F_b - 5800 = 26562.06 - 5800 = 20762.06 \, \text{N}\).

Step 4: Calculate the Mass of the Air Inside

The mass of the air inside the balloon is the weight of the air divided by the acceleration due to gravity: \( m_{\text{air}} = \frac{W_{\text{air}}}{g} = \frac{20762.06}{9.81} = 2116.62 \, \text{kg}\).

Step 5: Calculate the Average Density of the Heated Gases

Finally, the average density of the heated gases can be found using the formula \( \rho_{\text{gas}} = \frac{m_{\text{air}}}{V} = \frac{2116.62}{2200} = 0.9621 \, \text{kg/m}^3\). This is the average density of the heated gases inside the balloon envelope.

Key Concepts

Average Density Calculation

To calculate the average density of the heated gases inside a hot-air balloon, we need to begin by examining the concept of total lift. The air inside the balloon must be buoyant enough to lift not only the balloon itself, but also any additional weight it carries, like passengers and equipment.
The formula for average density is derived from the relationship between mass, volume, and density. We use:

• Total weight lifted (sum of all weights the balloon carries)
• Weight of the displaced air (buoyant force minus weight of all lifted content)

First, the total weight being lifted was calculated as 5800 N. By determining the weight of air inside with the formula \[W_{\text{air}} = F_b - \text{total lift} = 26562.06 \, \text{N} - 5800 \, \text{N} = 20762.06 \, \text{N}\]we next find the mass of the heated air: \[m_{\text{air}} = \frac{W_{\text{air}}}{g}\]. This formula allows us to transition from weight to mass, making it possible to calculate density, using the balloon's volume. Thus, the average density of the air is found by the formula: \[\rho_{\text{gas}} = \frac{m_{\text{air}}}{V}\].
This result provides insight into how light the heated gas becomes compared to the surrounding air.

Buoyant Force

Buoyant force is a fundamental concept in physics, particularly when dealing with floating or submerged objects. It refers to the force exerted by a fluid (like air or water) that supports the weight of an object.
For a hot-air balloon, this force determines how much the balloon can lift. The buoyant force can be calculated using the formula: \[ F_b = \rho_{\text{air}} \times g \times V \].
Where:

• \(\rho_{\text{air}}\) is the density of the surrounding air
• \(g\) is the acceleration due to gravity
• \(V\) is the volume of the balloon

This equation highlights that the lift (buoyant force) depends not only on the balloon's size (volume) but also on the density of the air it moves through.
When the calculated buoyant force exceeds the weight the balloon needs to lift, the balloon will rise. This principle keeps hot-air balloons afloat and allows them to carry passengers and cargo.

Heated Gases Density

The density of heated gases in a hot-air balloon plays a pivotal role in the balloon's ability to fly. By heating the air inside, the density decreases compared to the cooler air outside.
This difference in density is the key to generating lift. Heated air expands, takes up more space, and as a result, the same mass of air occupies a larger volume. This means it becomes less dense.
The challenge is achieving the right balance:

• If the density of the heated air becomes too low, the balloon won't rise beyond a certain altitude because it lacks support.
• If not heated enough, the air's density remains too high, preventing sufficient lift.

By calculating and adjusting the internal temperature of the air, balloon pilots achieve the optimal average density, ensuring a steady ascent and controlled flight. This is crucial for maintaining balance and safety during a journey. Understanding the dynamics of heated gas density equips pilots with the knowledge to make informed decisions about altitude and speed control.