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

Understand the Problem

We need to find the payload of a balloon. The balloon has a weight of 50 kg and is filled with 685.2 kg of helium gas. It displaces 5108 kg of air. According to the Archimedes principle, the buoyant force is equal to the weight of the displaced fluid, which in this case is air.

Determine the Buoyant Force

The buoyant force is equal to the weight of the displaced air, which is 5108 kg. This means the upward force that acts on the balloon due to the air is equal to the weight of the displaced air.

Calculate the Total Weight of the Balloon and Helium

The total weight of the balloon filled with helium is the sum of the weight of the balloon itself and the weight of the helium gas. So, the total weight is \(50\, \text{kg} + 685.2\, \text{kg} = 735.2\, \text{kg}.\)

Calculate the Payload

The payload is the weight that the balloon can carry. It is given by the difference between the weight of the air displaced by the balloon and the total weight of the balloon plus the helium.\[\text{Payload} = 5108\, \text{kg} - 735.2\, \text{kg} = 4372.8\, \text{kg}. \]

Final Step: Verify with Choices

From the calculation, the payload is 4372.8 kg, which corresponds to option (a).

Key concepts

Archimedes Principle

Archimedes' Principle is a fundamental law of physics that applies to fluid dynamics. It states that any object, fully or partially submerged in a fluid, experiences a buoyant force. This force is equal to the weight of the fluid that the object displaces. Archimedes' Principle explains why some objects float while others sink.
For a floating object like a balloon, understanding this principle helps determine how much load the balloon can carry. The force fighting gravity's pull on the balloon results from the weight of the air it displaces. This upward thrust allows the balloon to stay aloft and carry a payload. Knowing the total displaced fluid weight allows us to calculate how much extra weight (or payload) the balloon can support.

Buoyant Force

The buoyant force is the upward force exerted by a fluid that opposes the weight of an object immersed in it. This concept is key when calculating the payload capacity of a balloon filled with gas—a lightweight fluid that creates this force. The buoyant force acting on the balloon is found by measuring the volume of air its volume replaces.
To calculate it:

• Measure the fluid volume displaced by the object (in this case, the balloon's volume equals the fluid displaced).
• Multiply this volume by the fluid's density to find the buoyant force.

In this context, the buoyant force equals the weight of the displaced air, providing the lift that allows the balloon to float. If this force exceeds the total weight of the balloon and its contents, the balloon will rise and carry any additional load, or payload.

Displaced Fluid Weight

Understanding displaced fluid weight is crucial when determining buoyancy. It refers to the weight of the air (or any fluid) that a particular object displaces when submerged either partially or completely. In our balloon scenario, the balloon displaces a certain weight of air which determines how much additional weight the balloon can carry, also known as the payload.
When calculating the buoyant force, we consider the actual weight of the displaced air:

• For the balloon in question, it's given that it displaces 5108 kg of air.
• This displaced air weight becomes the direct measure of the buoyant force acting upon the balloon.

A clear understanding of displaced fluid weight helps solve many real-world applications, including determining which balloons would hold commercial payloads efficiently.

Helium Gas Properties

Helium gas is lighter than air, making it an excellent choice for inflating balloons intended to rise and float. Helium's key properties are its low density and non-reactive nature. The lighter weight of helium compared to air is what generates buoyancy, allowing balloons to lift off the ground.
Here are some important characteristics of helium used in buoyancy calculations:

• Low Density: Helium's density is significantly lower than that of air. When helium fills a balloon, it weighs less than the volume of air it displaces, which causes the balloon to rise.
• Inert and Non-Flammable: Helium is a noble gas, meaning it doesn’t react with other substances. This inertness makes helium safe for use in various applications, from party balloons to scientific experiments.

These properties make helium perfect for conditions requiring a non-combustible, light, and lifting medium, such as high-altitude balloon launches or scientific payload delivery systems.