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Chapter 13: Problem 46

A very large balloon with mass \(M=10.0 \mathrm{~kg}\) is inflated to a volume of \(20.0 \mathrm{~m}^{3}\) using a gas of density \(\rho_{\text {eas }}=\) \(0.20 \mathrm{~kg} / \mathrm{m}^{3}\). What is the maximum mass \(m\) that can be tied to the balloon using a \(2.00 \mathrm{~kg}\) piece of rope without the balloon falling to the ground? (Assume that the density of air is \(1.30 \mathrm{~kg} / \mathrm{m}^{3}\) and that the volume of the gas is equal to the volume of the inflated balloon).

Short Answer

Expert verified
The density of air is 1.30 kg/m³, and the volume of the balloon is 20.0 m³. Answer: To find the maximum mass that can be tied to the balloon, first compute the weight of the air displaced by the balloon using the formula: Weight of air = Density of air × Volume of air × acceleration due to gravity (g). Next, calculate the total weight of the balloon, rope, and lifted mass using the formula: Total weight = (Mass of balloon + Mass of rope + Mass of lifted mass) × g. Finally, equate the buoyant force to the total weight and solve for m. The maximum mass that can be tied to the balloon using a 2.00 kg piece of rope is approximately 25.0 kg.

Step by step solution

01

Compute the weight of the air displaced by the balloon

To find the buoyant force, first, calculate the weight of the air displaced by the balloon. The weight of the air can be calculated using the following formula: Weight of air = Density of air × Volume of air × acceleration due to gravity (g) Here, the density of air is given as 1.30 kg/m³, and the volume of the balloon is 20.0 m³. The acceleration due to gravity, g, can be taken as approximately 9.81 m/s². Weight of air = 1.30 kg/m³ × 20.0 m³ × 9.81 m/s² #Step 2: Calculate the total weight of the balloon, rope, and lifted mass#
02

Compute the total weight of the balloon, rope and lifted mass

To determine the maximum mass that can be tied to the balloon without it falling to the ground, we need to find the combined weight of the balloon, the rope, and this maximum mass. The total weight can be calculated with the following formula: Total weight = (Mass of balloon + Mass of rope + Mass of lifted mass) × g Given the mass of the balloon (M) is 10.0 kg and the mass of the rope is 2.00 kg. Let m be the maximum mass that can be lifted by the balloon. Then, we can write the total weight as: Total weight = (10.0 kg + 2.00 kg + m) × 9.81 m/s² #Step 3: Equate the buoyant force to the total weight and solve for m#
03

Equating the buoyant force to the total weight and solving for m

The buoyant force must be equal to the total weight for the balloon to remain afloat. Thus, we can write the following equation: Weight of air = Total weight Substitute the values from steps 1 and 2 into the equation and solve for m: 1.30 kg/m³ × 20.0 m³ × 9.81 m/s² = (10.0 kg + 2.00 kg + m) × 9.81 m/s² Solve the equation for m to find the maximum mass that can be tied to the balloon without it falling to the ground. Remember to include the mass of the rope in the final answer, as the problem asks for the maximum mass that can be tied to the balloon using a 2.00 kg piece of rope.

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