Density and Pressure Questions

In summary: The higher the density of something, the more force it has to push against something else to make it float. The balloon has to have more force than the surrounding air to keep it afloat.
  • #1
slowhands
2
0

Homework Statement



A hot air balloon has a volume of 1600m3. The balloon, its passengers, basket and other equipment have a mass of 500kg. Using the formula below, find the average temperature the air in the balloon must be raised to for it to lift off the ground.
Density of surrounding air = 1.3kgm-3.

Homework Equations



[tex]\rho=\rho_{0}\dfrac{273}{273+\Delta T}[/tex]

where [tex]\rho[/tex] = new density, [tex]\rho_{0}[/tex] = original density

The Attempt at a Solution



[tex]\rho=\dfrac{Mass}{volume}[/tex]
so
[tex]1.3=\dfrac{500}{1600}(\dfrac{273}{273+\Delta T})[/tex]

[tex]4.16=\dfrac{273}{273+\Delta T}[/tex]

[tex]1135.68 + 4.16\Delta T = 273[/tex]

[tex] \Delta T = -207.375[/tex]

But then this means the temperature change is negative which can't be right... Can anyone help please?
 
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  • #2
You tried to use 500/1600, which I assume was supposed to be 500kg/1600m^3, as a density. It's not. That's simply the mass of the basket and people divided by the volume of the balloon. To compute a density, you have to divide the mass of an object by the volume of the same object.

But anyway: that's not what the equation is meant to represent. The densities [itex]\rho[/itex] and [itex]\rho_0[/itex] are densities of air at two different temperatures. If you like, you can think of it this way: the product of density and temperature (in Kelvins) for air is a constant. So the value of that product at 273K,
[tex]273\mathrm{K}*\rho_0[/tex]
is equal to the value at some higher temperature [itex]273\mathrm{K}+\Delta T[/itex],
[tex]273\mathrm{K}*\rho_0 = (273\mathrm{K}+\Delta T) * \rho[/tex]

In this problem, you're trying to find [itex]\Delta T[/itex]. That means you'll need to get the other variables in the equation from other sources. [itex]\rho_0[/itex] is the density of air at [itex]273\mathrm{K}[/itex], which you're given. [itex]\rho[/itex] is the density of air at the higher temperature. It needs to be the right density so that the balloon as a whole will float. How would you figure out what that is?
 
  • #3
The new density of the air in the balloon has to be less than or equal to the surrounding air for the balloon to float, right? But I don't know how you could work out how much less than 1.3kgm-3 it has to be... Would you need to use

[tex] V=V_o(1+\dfrac{\Delta T}{273})[/tex]

We were given this in a previous question
 
  • #4
Not less than or equal to - strictly less than, by at least some amount. Remember that the balloon has to float despite the weight of the basket and people (and the balloon itself) pulling it down. If the density of the air in the balloon were equal to the density of the outside air, how could it float? (It couldn't)

Surely you're familiar with the concept of buoyancy? Buoyant force?
 

Related to Density and Pressure Questions

1. What is the difference between density and pressure?

Density is the measure of how much mass is contained in a given volume of a substance. Pressure, on the other hand, is the force exerted per unit area. In simpler terms, density describes how compact a substance is, while pressure describes the force that the substance exerts on its surroundings.

2. How are density and pressure related?

Density and pressure are directly proportional to each other, meaning that as density increases, so does pressure. This is because the more compact a substance is, the more particles are in a given space, resulting in a greater force being exerted on the surrounding area.

3. How is density measured?

Density is typically measured in units of mass per unit volume, such as grams per cubic centimeter (g/cm3) or kilograms per cubic meter (kg/m3). It can be measured using various tools, such as a balance and a graduated cylinder, to determine the mass and volume of a substance.

4. How does pressure change with depth?

As depth increases, the weight of the fluid above an object also increases, resulting in an increase in pressure. This is known as hydrostatic pressure and is one of the reasons why the pressure in the ocean increases with depth. For every 10 meters of depth, the pressure increases by about 1 atmosphere.

5. How does temperature affect density and pressure?

Temperature can have a significant impact on both density and pressure. As temperature increases, the particles in a substance gain more energy and move faster, causing them to spread out and decrease in density. This decrease in density also results in a decrease in pressure, as there are fewer particles exerting a force on a given area.

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