Question

A hot-air balloon can lift a weight of \(5626 \mathrm{~N}\) (including its own weight). The density of the air outside the balloon is \(1.205 \mathrm{~kg} / \mathrm{m}^{3}\). The density of the hot air inside the balloon is \(0.9449 \mathrm{~kg} / \mathrm{m}^{3}\). What is the volume of the balloon?

Short answer

Answer: The approximate volume of the hot air balloon is 2202 cubic meters.

Step-by-step solution

Calculate the mass of the hot air balloon

To find the mass of the hot air balloon, we can use the weight-lifting capacity of the balloon (including its own weight) and the acceleration due to gravity. The formula to find the mass of the hot air balloon is: \(m_{balloon} = \frac{W_{balloon}}{g}\) Where: \(m_{balloon}\): mass of the hot air balloon (kg) \(W_{balloon}\): weight of the hot air balloon (N) \(g\): acceleration due to gravity (\(9.81 \mathrm{~m/s^2}\)) Now we can plug in the given values: \(m_{balloon} = \frac{5626 \mathrm{~N}}{9.81 \mathrm{~m/s^2}} = 573.1 \mathrm{~kg}\)

Apply Archimedes' principle to find the balloon's volume

Archimedes' principle states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. In this case, the buoyant force on the hot air balloon is equal to the weight of the air displaced by the balloon minus the weight of the hot air inside the balloon. The buoyant force (\(F_{buoyant}\)) can be calculated using the following equation: \(F_{buoyant} = W_{air} - W_{hotair}\) where: \(W_{air}\): weight of the air displaced by the balloon \(W_{hotair}\): weight of the hot air inside the balloon Since \(F_{buoyant} = W_{balloon}\), we can say: \(W_{balloon} = W_{air} - W_{hotair}\) We can convert the weight of the air and hot air to mass, and use their densities to find the volume of the balloon. \(W_{air} = m_{air}g = V_{balloon}ρ_{air}g\) \(W_{hotair} = m_{hotair}g = V_{balloon}ρ_{hotair}g\) Now, we can substitute these equations into the buoyant force equation: \(W_{balloon} = (V_{balloon}ρ_{air}g) - (V_{balloon}ρ_{hotair}g)\) We can simplify this equation by cancelling out the common term \(g\): \(m_{balloon} = V_{balloon}(ρ_{air} - ρ_{hotair})\) Now, rearrange the equation to find the volume of the balloon (\(V_{balloon}\)): \(V_{balloon} = \frac{m_{balloon}}{(ρ_{air} - ρ_{hotair})}\)

Calculate the volume of the balloon

Now we can plug in the known values for the mass of the balloon and the densities of the air and hot air: \(V_{balloon} = \frac{573.1 \mathrm{~kg}}{(1.205 \mathrm{~kg/m^3} - 0.9449 \mathrm{~kg/m^3})}\) \(V_{balloon} = \frac{573.1 \mathrm{~kg}}{0.2601 \mathrm{~kg/m^3}}\) \(V_{balloon} \approx 2202 \mathrm{~m^3}\) The volume of the hot air balloon is approximately 2202 cubic meters.

Key concepts

Archimedes' Principle and Its Application

Archimedes' principle is a fundamental concept in physics that explains why objects float or sink in a fluid. It states that an object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object. This principle applies not only to objects submerged in water but also to those in air, like a hot air balloon.

When considering a hot air balloon, Archimedes' principle helps us understand how the balloon rises. The balloon displaces a certain volume of air as it inflates. The weight of this displaced air generates a buoyant force. If the buoyant force is greater than the weight of the balloon (including the hot air inside, its structure, and any additional weight it carries), the balloon will ascend. By adjusting the density of the air inside the balloon, usually by heating it, the balloon's overall density becomes less than that of the cooler surrounding air, leading to an increase in the buoyant force that can lift the balloon off the ground.

In the exercise, the application of Archimedes' principle is critical to calculate the volume of the balloon. The problem provides the necessary forces and densities, asking us to compute the balloon's volume using the principle. It shows how deeply intertwined this principle is with the basic mechanics of how a hot air balloon operates.

Buoyant Force in Action

The buoyant force is the uplifting force experienced by an object when it displaces a fluid. In the context of our hot air balloon, the fluid is the air. This force is directly related to Archimedes' principle. However, understanding the dynamics of the buoyant force requires knowing both the volume of the object and the density of the fluid it displaces. In our exercise, the buoyant force is what counteracts the weight of the hot air balloon, enabling it to float.

The formula introduced in the solution, where the buoyant force is equal to the difference between the weight of the air displaced and the weight of the hot air, simplifies the underlying physics of a hot air balloon's flight. It's a prime example of the buoyant force in a real-world scenario, illustrating the conditions under which an object remains afloat in a fluid or, in this case, air. Students often benefit from real-world applications like this, helping them visualize and understand the concepts at play. The hot air balloon's ability to lift is entirely based on its ability to create a sufficient buoyant force.

Density and Volume Relationship

Density and volume are two concepts crucial to understanding the physics behind a hot air balloon's buoyancy. Density is defined as an object's mass per unit volume. For our problem, we deal with the density of the air inside the balloon compared to the air outside. A hot air balloon utilizes the principle that warmer air is less dense than cooler air. By heating the air inside the balloon, its density decreases.

When the volume enters the picture, things get interesting. The volume of the balloon determines how much air it displaces. A larger balloon will displace more air and can create a greater buoyant force, assuming the density of the air inside it is sufficiently reduced. The exercise asked us to find the volume based on the weight the balloon can lift and the respective densities of the air inside and outside the balloon—highlighting the dependency between volume and density in determining the balloon’s potential to rise. Remember, a hot air balloon's volume isn't just about size; it's a crucial part of the calculation that allows it to become airborne through the interplay with density.