Calculating Diameter of Balloon to Lift 20 Metric Tons

[xenoflauge]

Homework Statement

balloons are required to lift 20 metric tons. the pressure of helium is 101.325kpa. temp=21.1degcelsius.

what should be the diameter of the balloon if it was spherical in shape in order to lift the weight?

Homework Equations

bouyant force??

The Attempt at a Solution

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tnx.

 

[berkeman]

You need to show more work in order for us to help you. Tell us more about bouyant forces. What causes them? If you have an object that normally floats in water, and you hold it under water, how do you calculate the upward force that you feel from the object?

 

[piercas]

To calculate the diameter of the balloon needed to lift 20 metric tons, we first need to determine the volume of helium required. This can be done using the ideal gas law: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature. We are given the pressure (101.325 kPa) and temperature (21.1 degrees Celsius) of the helium, and we know that the volume of the balloon will be the same as the volume of the helium inside it. We can also assume that the number of moles of helium will be the same as the number of moles of air displaced by the weight of the balloon.

Next, we need to determine the buoyant force acting on the balloon. This is the force that will counteract the weight of the balloon and its cargo. It can be calculated using the formula Fb = ρVg, where ρ is the density of the surrounding air, V is the volume of the balloon, and g is the acceleration due to gravity. We know the density of air at sea level (1.225 kg/m^3) and we can calculate the volume of the balloon using the ideal gas law.

Once we have the buoyant force and the weight of the cargo (20 metric tons), we can use Newton's second law (F = ma) to find the acceleration of the balloon. We can then use this acceleration to determine the height the balloon will need to reach in order to lift the cargo. Finally, using the formula for the volume of a sphere (V = 4/3πr^3), we can calculate the diameter of the balloon needed to reach this height.

It is important to note that this calculation is based on certain assumptions and may not be entirely accurate. Other factors, such as wind resistance and the strength of the balloon material, may also affect the diameter needed. It is always important to conduct further research and testing before attempting to lift such a heavy load with a balloon.

 

[piercas]

To calculate the diameter of the balloon, we first need to determine the volume of the balloon needed to lift 20 metric tons. This can be done by using the buoyant force equation, which is equal to the weight of the fluid displaced by the object. In this case, the fluid is the air and the object is the balloon. We can use the following equation:

Buoyant force = (density of air) x (volume of balloon) x (acceleration due to gravity)

Since we know the weight of the object (20 metric tons or 20,000 kg) and the acceleration due to gravity (9.8 m/s^2), we can rearrange the equation to solve for the volume of the balloon:

Volume of balloon = (weight of object) / (density of air x acceleration due to gravity)

Next, we need to determine the density of air at the given temperature and pressure. This can be done using the ideal gas law, which states that the pressure of a gas is equal to its density times its temperature and a constant value. We can rearrange the equation to solve for the density of air:

Density of air = (pressure of helium) / (constant value x temperature)

Now, we can plug in the given values and solve for the density of air:

Density of air = (101.325 kPa) / (8.314 J/mol x 294.25 K) = 0.0403 kg/m^3

Finally, we can plug this value into the equation for the volume of the balloon and solve for the diameter:

Volume of balloon = (20,000 kg) / (0.0403 kg/m^3 x 9.8 m/s^2) = 51,709 m^3

Volume of a sphere = (4/3) x π x (radius)^3

Solving for the radius, we get:

Radius = 21.02 m

Therefore, the diameter of the spherical balloon needed to lift 20 metric tons in these conditions would be approximately 42.04 m. It is important to note that this calculation does not take into account the weight of the balloon itself, which would need to be factored in for a more accurate calculation.