Problem 37 - College Physics

Question

On October 21, 2001, Ian Ashpole of the United Kingdom achieved a record altitude of \(3.35 \mathrm{~km}\) (11 \(000 \mathrm{ft}\) ) powered by 600 toy balloons filled with helium. Each filled balloon had a radius of about \(0.50 \mathrm{~m}\) and an estimated mass of \(0.30 \mathrm{~kg}\). (a) Estimate the total buoyant force on the 600 balloons. (b) Estimate the net upward force on all 600 balloons. (c) Ashpole parachuted to Earth after the balloons began to burst at the high altitude and the system lost buoyancy. Why did the balloons burst?

Step-by-Step Solution

Answer

The total buoyant force on the balloons and net upward force on them can be calculated using Archimedes' principle and Newton’s second law of motion respectively. The balloons burst at high altitude due to the pressure differential between the inside and the outside of the balloon, which becomes too great for the balloon material to withstand.

Step 1: Calculate the Volume of a Single Balloon

First, find the volume of a single balloon using the formula for the volume of a sphere, which is given by \(V = \frac{4}{3}\pi r^3\), where \(r\) is the radius. Given that each balloon has a radius of 0.50 m, the volume can be calculated as: \(V = \frac{4}{3}\pi (0.50 m)^3\).

Step 2: Calculate the Total Buoyant Force on the Balloons

Next, calculate the total buoyant force that acts on all the balloons. This can be done using Archimedes' principle, which indicates that the buoyant force is equal to the weight of the fluid (air here) displaced by the balloons. The formula to use here is \(F_{b} = \rho_{fluid} \times V \times g \times n\), where \(F_{b}\) is the buoyant force, \(\rho_{fluid}\) is the density of air, \(V\) is the volume of a balloon, \(g\) is the acceleration due to gravity, and \(n\) is the number of balloons. Be aware that the density of air decreases as altitude increases. However, for simplicity, use the standard sea-level value: \(\rho_{fluid} = 1.225 kg/m^3\). Then compute \(F_{b} = 1.225 kg/m^3 \times V \times 9.8 m/s^2 \times 600\).

Step 3: Calculate the Total Weight of the Balloons

Calculate the total weight of the 600 balloons. This is given by the formula \(F_{g} = m \times g \times n\), where \(F_{g}\) is the gravitational force (weight), \(m\) is the mass of a single balloon, \(g\) is the acceleration due to gravity, and \(n\) is the number of balloons. Considering that the mass of each balloon is 0.30 kg, the total weight can be calculated by \(F_{g} = 0.30 kg \times 9.8 m/s^2 \times 600\).

Step 4: Calculate the Net Upward Force on the Balloons

Now, calculate the net upward force on all 600 balloons. This can be found by subtracting the total weight of the balloons from the total buoyant force, that is \(F_{net} = F_{b} - F_{g}\).

Step 5: Answer Why the Balloons Burst at High Altitudes

Finally, explain why the balloons burst at high altitude. As altitude increases, external air pressure decreases but the pressure inside the balloons remains the same. This creates a greater pressure differential, causing the balloon to expand until the balloon material can no longer withstand the tension and thus bursts.

Key Concepts

Archimedes' Principle

Archimedes' Principle is a fundamental concept in fluid dynamics that describes how objects immersed in a fluid experience a buoyant force. This principle states that the buoyant force is equal to the weight of the fluid displaced by the object.

An easy way to understand this is to think of an object submerged in water. As it displaces water, the object experiences an upward force. This force is what we call buoyancy. In our balloon scenario, each helium-filled balloon displaces a certain volume of air, which contributes to the overall buoyant force.

The principle can be mathematically represented by: \[ F_b = \rho_{fluid} \times V \times g \] where

• \( F_b \) is the buoyant force,
• \( \rho_{fluid} \) is the fluid density,
• \( V \) is the displaced volume, and
• \( g \) is the acceleration due to gravity.

It's essential to know that the greater the volume of air displaced, the higher the buoyant force.

Volume of a Sphere

The volume of a sphere is a crucial calculation when examining spherical objects like balloons. This is because it helps us determine how much fluid (air) an object will displace, which in turn affects the buoyant force.

The formula for calculating the volume of a sphere is: \[ V = \frac{4}{3} \pi r^3 \] where

• \( V \) is the volume,
• \( \pi \) is a constant roughly equal to 3.14159,
• \( r \) is the radius of the sphere.

In the case of a toy balloon with a radius of 0.50 meters, this formula lets us compute the volume, which is then used to find out how much air it displaces. This volume is essential for calculating the buoyant force using Archimedes' principle. Understanding how volume influences buoyancy allows us to predict how much weight a balloon can lift and the extent of its ascendancy into the atmosphere.

Pressure Differential

Pressure differential is the difference in pressure between two points. In the context of balloons, it refers to the difference between the pressure inside the balloon and the external atmospheric pressure.

Pressure differential plays a significant role in why balloons burst at high altitudes. As a balloon ascends, the external air pressure decreases, while the inner pressure remains constant. This causes the balloon's material to experience more tension.

When the internal pressure far exceeds the external pressure, the material of the balloon stretches and may eventually reach a breaking point. The greater the altitude, the larger the differential, leading towards the bursting of the balloon. This understanding is vital when determining how high a balloon can rise without popping.

Altitude Effects on Air Pressure

Altitude has a direct impact on air pressure. As one moves to higher altitudes, the atmospheric pressure decreases. This is because there is less air above you, thus fewer air molecules to exert pressure downwards.

This connects back to our principle of buoyancy and why Ian Ashpole's balloons eventually burst. At sea level, air pressure is at its highest. But as the balloons ascend to 3.35 kilometers, the pressure is substantially lower.

To visualize this: at lower altitudes, the density of air remains high. This gives a constant buoyant force if the air density value used is from sea level. However, as one moves higher:

• The atmospheric pressure decreases,
• Drastic pressure differences arise between the inside and outside of the balloon,
• This results in the expansion and potential rupture of the balloon.

Understanding these altitude effects is crucial not only for balloons but in any application involving gases and fluid dynamics, including aviation and meteorology.