Question

A balloon having weight \(50 \mathrm{~kg}\) is filled with \(685.2 \mathrm{~kg}\) of helium gas at \(760 \mathrm{~mm}\) pressure and \(25^{\circ} \mathrm{C}\). What will be its pay load if it displaces \(5108 \mathrm{~kg}\) of air? (a) \(4372.8 \mathrm{~kg}\) (b) \(4392.6 \mathrm{~kg}\) (c) \(4444.4 \mathrm{~kg}\) (d) \(3482.9 \mathrm{~kg}\)

Short answer

The pay load of the balloon is 4372.8 kg (option a).

Step-by-step solution

Calculate Total Lift

The balloon displaces 5108 kg of air, which means it can lift this amount in terms of displacement weight. We start with this maximum upward lift.

Subtract the Balloon's Weight

First, we subtract the weight of the balloon itself from the total displaced weight. The weight of the balloon is 50 kg, leaving us with: \[ 5108\, \text{kg} - 50\, \text{kg} = 5058\, \text{kg} \]

Subtract the Helium's Weight

Next, subtract the weight of the helium gas inside the balloon, which is 685.2 kg: \[ 5058\, \text{kg} - 685.2\, \text{kg} = 4372.8\, \text{kg} \]

Determine the Payload

The result from the previous step shows that the remaining capacity after accounting for the helium and balloon weight is the payload the balloon can carry. Thus, the payload capacity equals 4372.8 kg.

Key concepts

Gas Laws

Understanding gas laws is fundamental when dealing with gases in various conditions. The gas laws describe how gases behave, mainly in relation to pressure, volume, and temperature. The principal gas laws include Boyle's Law, Charles's Law, and Avogadro's Law. These laws come together in the Ideal Gas Law, which is summarized by the equation: \[ PV = nRT \] where:

• P stands for pressure.
• V represents volume.
• n is the number of moles of gas.
• R is the gas constant.
• T is the temperature in Kelvin.

This formula allows us to predict how a gas will behave when conditions change, such as in a balloon expanding against atmospheric pressure. For the helium in our balloon, using these principles can precisely determine its behavior under given conditions. The gases fill the balloon and exert an outward pressure, counterbalancing the atmospheric pressure outside. As the gas laws illustrate, the balloon will maintain equilibrium until external factors like temperature or pressure change.

Buoyancy in Gases

Buoyancy is a force exerted by a fluid (like air) that opposes an object's weight. It depends on the displaced fluid's weight according to Archimedes' principle. This principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid it displaces. In simple terms:

• An object will float if the buoyant force is equal to its weight.
• An object will sink if its weight is greater than the buoyant force.
• An object remains neutrally buoyant if the two forces are equal.

Applying this to our balloon filled with helium, it displaces air when it rises. The balloon weighs 50 kg and displaces 5108 kg of air. Since the weight of the displaced air is greater than the total weight of the balloon and the helium inside, the balloon experiences an upward buoyant force large enough to lift it and any additional payload on board. It’s this principle that allows the balloon to ascend carrying a load.

Weight and Mass Calculations

Weight and mass are related but distinct concepts. Mass refers to the amount of matter in an object and is measured in kilograms (kg). Weight, however, is the force exerted by gravity on that mass. It is calculated using the equation: \[ W = mg \] where:

• W is the weight.
• m is the mass.
• g is the acceleration due to gravity, usually \(9.81 \, \text{m/s}^2\).

In the problem, we determine the weight of the balloon itself and the helium it contains in order to calculate the net lift capacity. The calculations began by subtracting these weights from the displaced air's lifted weight, leaving us with the payload a balloon can transport without losing lift. Understanding this distinction is crucial when performing calculations related to buoyancy and lift, as weight is the factor counterbalanced by the buoyant force.