Question

A hot-air balloon has a volume of \(2435 \mathrm{~m}^{3}\). The balloon can lift a weight of \(6194 \mathrm{~N}\) (including its own weight). The density of the air outside the balloon is \(1.205 \mathrm{~kg} / \mathrm{m}^{3}\). What is the density of the hot air inside the balloon?

Short answer

Answer: The approximate density of the hot air inside the balloon is 0.9446 kg/m³.

Step-by-step solution

Understanding buoyant force

According to Archimedes' principle, the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. In this case, the buoyant force acting on the hot-air balloon is equal to the difference in the weight of the air inside the balloon and the weight of the air it displaces.

Calculate the mass of the air displaced

To find the mass of the air displaced, we will use the given volume of the balloon and the density of the air outside the balloon. The formula for the mass is: Mass = Density * Volume Using the given values, Mass = 1.205 kg/m³ * 2435 m³ Mass ≈ 2934.175 kg

Calculate the weight of the air displaced

To calculate the weight of the air displaced, we will use the formula: Weight = Mass * Gravity Using the standard acceleration due to gravity (g = 9.81 m/s²), Weight ≈ 2934.175 kg * 9.81 m/s² Weight ≈ 28766.65775 N

Calculate the weight of the hot air inside the balloon

We know that the balloon can lift a total weight of 6194 N. Therefore, the weight of the hot air inside the balloon is equal to the difference between the weight of the air displaced and the weight it can lift: Weight of hot air = Weight of air displaced - Weight it can lift Weight of hot air ≈ 28766.65775 N - 6194 N Weight of hot air ≈ 22572.65775 N

Calculate the mass of the hot air inside the balloon

Now that we have the weight of the hot air inside the balloon, we can calculate its mass using the formula: Mass = Weight / Gravity Using the standard acceleration due to gravity (g = 9.81 m/s²), Mass ≈ 22572.65775 N / 9.81 m/s² Mass ≈ 2300.175 kg

Calculate the density of the hot air inside the balloon

Lastly, we will use the mass of the hot air inside the balloon and the volume of the balloon to calculate the density of the hot air. The formula for density is: Density = Mass / Volume Using the given volume of the balloon (2435 m³) and the calculated mass of the hot air, Density ≈ 2300.175 kg / 2435 m³ Density ≈ 0.9446 kg/m³ The density of the hot air inside the balloon is approximately 0.9446 kg/m³.

Key concepts

Archimedes' Principle

Archimedes' principle is a fundamental scientific concept that explains why objects like hot-air balloons are able to float in the air. The principle states that an object submerged in a fluid is buoyed up by a force that is equal to the weight of the fluid displaced by the object.

In practical terms, when a hot-air balloon is filled with hot air, it displaces a certain volume of the cooler, denser air surrounding it. The force pushing the balloon upwards—the buoyant force—is equal to the weight of this displaced air.

To solve a hot-air balloon problem using Archimedes' principle, we need to understand that if the balloon floats, the buoyant force must balance the weight of the balloon itself and the weight of the hot air inside. This concept is crucial when calculating how high a balloon can ascend or determining what density the hot air inside needs to be for the balloon to lift off.

Buoyant Force

The buoyant force comes into play as part of Archimedes' principle and is vital in solving problems related to hot-air balloons. It can be thought of as the fluid's pushback against the weight of the object submerged in it. For our hot-air balloon, the buoyant force must be equal to the balloon's weight if it is to hover at a constant altitude.

In the step-by-step solution for the hot-air balloon problem, you'll note that the buoyant force is the key force that allows the balloon to lift. It is the difference between the weight of the air displaced by the balloon and the weight of the hot air within it.

Understanding how to calculate this force is essential because it affects everything from the balloon's lift capacity to its stability in flight. To increase the balloon's altitude, you would need to decrease the density of the hot air inside it, effectively increasing the buoyant force.

Density Calculation

Density calculation is fundamental in understanding the physics behind hot-air balloons. The density of the air inside the balloon, compared to the air outside, dictates whether the balloon will sink, float, or rise.

Density is defined as mass per unit volume, expressed in the formula:
\( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). For a hot-air balloon, we are essentially comparing the density of the hot air inside the balloon (which is less dense due to heating and expansion) with the colder, denser air outside.

To calculate the density of the hot air, we need the mass of the air inside the balloon, which we derive from the weight (considering standard gravitational force) and the volume of the balloon, as shown in the steps of the solution. Recognizing how density affects the buoyant force and the balloon's lifting power is key. The lesser the density of the air inside the balloon, the greater the lift it can achieve.