**Problem 6: Buoyancy and Lifting Capacity of a Spherical Helium Balloon**A spherically shaped balloon has a radius of 13.5 meters and is filled with helium. Determine the additional mass a cargo can lift, provided that the combined mass of the balloon's skin and cargo is 900 kilograms. For this calculation, neglect the buoyant force acting on the cargo's volume itself. - The density of air is given as 1.29 kg/m³.- The density of helium is 0.179 kg/m³.**Solution:**To find the extra mass that can be lifted, calculate the buoyant force and subtract the mass of the helium and balloon structure. Let's assume that `V` is the volume of the balloon, `ρ_air` is the density of air, and `ρ_helium` is the density of helium. 1. Calculate the volume of the balloon using the formula for the volume of a sphere: \[ V = \frac{4}{3} \pi r^3 \] where \( r = 13.5 \) meters.2. Use the volume to determine the buoyant force: \[ \text{Buoyant Force} = V \times \rho_{\text{air}} \times g \] where \( g \approx 9.81 \, \text{m/s}^2 \).3. Calculate the weight of the helium in the balloon: \[ \text{Weight of Helium} = V \times \rho_{\text{helium}} \times g \]4. Determine the lifting capacity by subtracting the mass of the balloon and helium from the buoyant force and include the initial mass: \[ \text{Extra Mass Lifted} = \left(\frac{\text{Buoyant Force} - \text{Weight of Helium}}{g}\right) - \text{Mass of Balloon+Cargo} \] Round off your answer to the nearest kilogram.

Calculate the volume of the spherical balloon. V=43πr3=43π13.53=10,306 m3 Therefore, the mass of helium: mHe=ρV=0.179×10,306=1844.774 kg Now, the total mass will have its weight equal to the buoyant force exerted on the balloon if the balloon is to lift the total mass. ρairV×g=mtotal gρairV=mtotal 1.29×10306=mtotalmtotal=13,294.74 kg Also, the expression of total mass: mtotal =mHe+mextra+90013,294.74=1844.774+mextra+900mextra=10,549.966 kg Hence, the extra mass is 10,549.966 kg.

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6. A spherically shaped balloon has a radius of 13.5 m, and is filled with helium. How much extra mass can a cargo lift, assuming that the skin and the cargo of the balloon have a mass of 900 kg? Neglect the buoyant force on the cargo volume itself. (The density of air is 1.29 kg/m3 and the density of helium is 0.179 kg/m³)

6. A spherically shaped balloon has a radius of 13.5 m, and is filled with helium. How much extra mass can a cargo lift, assuming that the skin and the cargo of the balloon have a mass of 900 kg? Neglect the buoyant force on the cargo volume itself. (The density of air is 1.29 kg/m3 and the density of helium is 0.179 kg/m³)

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

**Problem 6: Buoyancy and Lifting Capacity of a Spherical Helium Balloon**

A spherically shaped balloon has a radius of 13.5 meters and is filled with helium. Determine the additional mass a cargo can lift, provided that the combined mass of the balloon's skin and cargo is 900 kilograms. For this calculation, neglect the buoyant force acting on the cargo's volume itself.

- The density of air is given as 1.29 kg/m³.
- The density of helium is 0.179 kg/m³.

**Solution:**

To find the extra mass that can be lifted, calculate the buoyant force and subtract the mass of the helium and balloon structure. Let's assume that `V` is the volume of the balloon, `ρ_air` is the density of air, and `ρ_helium` is the density of helium.

1. Calculate the volume of the balloon using the formula for the volume of a sphere:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r = 13.5 \) meters.

2. Use the volume to determine the buoyant force:
\[
\text{Buoyant Force} = V \times \rho_{\text{air}} \times g
\]
where \( g \approx 9.81 \, \text{m/s}^2 \).

3. Calculate the weight of the helium in the balloon:
\[
\text{Weight of Helium} = V \times \rho_{\text{helium}} \times g
\]

4. Determine the lifting capacity by subtracting the mass of the balloon and helium from the buoyant force and include the initial mass:
\[
\text{Extra Mass Lifted} = \left(\frac{\text{Buoyant Force} - \text{Weight of Helium}}{g}\right) - \text{Mass of Balloon+Cargo}
\]

Round off your answer to the nearest kilogram.

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