A hot-air balloon consists of a basket hanging beneath a...

A hot-air balloon consists of a basket hanging beneath a large envelope filled with hot air. A typical hot-air balloon has a total mass of 545 $\mathrm{kg}$ , including passengers in its basket, and holds $2.55 \times 10^{3} \mathrm{m}^{3}$ of hot air in its envelope. If the ambient air density is 1.25 $\mathrm{kg} / \mathrm{m}^{3}$ , determine the density of hot air inside the envelope when the balloon is neutrally buoyant. Neglect the volume of air displaced by the basket and passengers.

Key Concepts

Mass-Volume-Density Relationship

The relationship between mass, volume, and density (expressed as density = mass/volume) is fundamental in calculating how much fluid an object displaces and, consequently, the buoyant force acting on it. This understanding is essential for solving problems involving equilibrium in fluids, such as determining the necessary internal density of a gas to achieve flotation.

Archimedes' Principle

This principle states that any object immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the object. It is the foundational concept for understanding how and why objects float, and it applies universally to any scenario involving fluids and submerged or partially submerged bodies.

Neutral Buoyancy

Neutral buoyancy occurs when the buoyant force acting on an object exactly equals the object's weight, resulting in no net force causing the object to ascend or descend. This concept is crucial in fields such as aerostatics and underwater operations, where maintaining a constant altitude or depth is desired.

Density

Density is a measure of mass per unit volume and plays a central role in determining whether an object will float or sink in a fluid. Comparing the density of an object or gas to that of the surrounding fluid allows one to predict the direction and magnitude of the buoyant force.

Transcript

00:01 In the given problem, suppose here this is the hot air balloon and this is the basket attached with it having some passengers in it.

00:20 Volume of the balloon is supposed represented as vb.

00:31 The density of hot air is represented as row, h .a, mass.

00:45 Of the passenger with basket that is m and density of air is suppose row a.

01:09 Now the mass of the basket along with the passenger will be giving this weight mg acting downward vertically downward the weight of this balloon that will also be acting vertically downward, which is given by the volume of this balloon means v b into the density of hot air inside it, row h .a into g.

01:41 And as the balloon is having no external, no net force acting on it, so as per arquodiz principle, the point force will be acting on it in upward and direction and that buoy force will be equal to the addition of these two forces acting vertically downward.

02:10 Hence we can say vb into row h a into g plus m g is equal to v b for point force that is the volume that is the weight of the air displaced so that will be volume of the balloon because the same equivalent amount of volume of air will be displaced by the balloon.

02:36 So this is vb, the volume of the air displaced into density of air into g.

02:43 Then this g can be taken as a common out.

02:46 So leaving behind vb into row h a plus m is equal to vb into row a into g.

02:57 So canceling this g from both the sides...

Please give Ace some feedback