You want to design a helium-filled balloon that will lift a total payload of 1225 kg. What volume of helium is needed to just barely lift this payload?
Use Archimedes' Principle.
Volume*(Air density) = Payload + (Volume)*(Helium density)
Solve for Volume.
You will need to assume a temperature and pressure. I suggest 15 C and 1 atmosphere.
At sea level and at 15°C , the density of air is 1.275 kg/m^3. Helium would be 4/29 of that, or 0.176 kg/m^3
V = 1225 kg/(1.275 - 0.176)kg/m^3
= 1115 m^3
The "payload" is assumed to include the weight of all balloon materials, not just cargo
To calculate the volume of helium needed to lift a payload of 1225 kg, you need to consider the buoyancy force of the balloon.
The buoyancy force (Fb) can be calculated using Archimedes' principle:
Fb = Volume of helium * Density of air * Acceleration due to gravity
Where:
- Volume of helium is the volume of the balloon
- Density of air is the density of air at the given conditions
- Acceleration due to gravity is approximately 9.8 m/s^2
For the balloon to just barely lift the payload, the buoyancy force must equal the weight of the payload (Fg):
Fg = Mass of payload * Acceleration due to gravity
Now we can set these equations equal to each other and solve for the volume of helium:
Volume of helium * Density of air * Acceleration due to gravity = Mass of payload * Acceleration due to gravity
Volume of helium = (Mass of payload * Acceleration due to gravity) / (Density of air * Acceleration due to gravity)
As the Acceleration due to gravity cancels out, the equation simplifies to:
Volume of helium = Mass of payload / Density of air
Now, we can substitute the given values:
Mass of payload = 1225 kg
Density of air at sea level and room temperature (25 degrees Celsius) is approximately 1.225 kg/m^3.
Using this information, let's calculate the volume of helium needed:
Volume of helium = 1225 kg / 1.225 kg/m^3
Volume of helium = 1000 m^3
Therefore, to just barely lift a payload of 1225 kg, you would need a volume of helium equal to 1000 cubic meters.
To find the volume of helium needed to lift the payload, we need to consider the buoyant force acting on the balloon. The buoyant force is equal to the weight of the air displaced by the balloon, and it acts in the opposite direction of gravity.
To calculate the volume of helium needed, we can use Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by an object.
Step 1: Determine the weight of the payload
The weight of the payload is given as 1225 kg. We can use the equation W = mg, where W is the weight, m is the mass, and g is the acceleration due to gravity. Assuming g = 9.8 m/s^2, we can calculate the weight of the payload:
W_payload = 1225 kg * 9.8 m/s^2
= 11,985 N
Step 2: Calculate the weight of the displaced air
The weight of the displaced air is equal to the buoyant force acting on the balloon. Since the buoyant force and weight are equal in magnitude but opposite in direction, the weight of the displaced air is 11,985 N.
Step 3: Determine the volume of the helium
The volume of the helium can be calculated using the equation:
V_helium = weight of the displaced air / density of air
The density of air at sea level and room temperature is approximately 1.225 kg/m^3.
V_helium = 11,985 N / 1.225 kg/m^3
≈ 9786.94 m^3
Therefore, to just barely lift a payload of 1225 kg, you would need a volume of approximately 9786.94 cubic meters of helium in the balloon.