A 600-kg weather ballon is designed to lift a 4000-kg package. What volume should the ballon have after being inflated with helium at standard temperature and pressure so the total load can be lifted up?
The problem asks for volume. Take it a step further...
assume the ballon is spherical, and find it's radius.
Ok so I know how to find the volume. I just need to know how to get the radius.
Since V = (4/3)*pi*R^3 (for a sphere),
R = cube root of [3V/(4*pi)]
Thank you! Took me a little whilez to figure it out but I got it.
To find the volume of the balloon and its radius, we can start by using the concept of buoyancy.
Buoyancy is the upward force exerted on an object submerged in a fluid (in this case, helium) and is equal to the weight of the fluid displaced by the object. In this case, the balloon will displace air, so the weight of the air it displaces should be equal to the combined weight of the weather balloon and the package.
Given that the weather balloon has a mass of 600 kg and the package has a mass of 4000 kg, the total weight is:
Total weight = weight of the balloon + weight of the package
Total weight = (mass of the balloon * acceleration due to gravity) + (mass of the package * acceleration due to gravity)
Total weight = (600 kg * 9.8 m/s^2) + (4000 kg * 9.8 m/s^2)
Next, we need to find the weight of the displaced air, which is equal to the buoyant force:
Buoyant force = weight of the displaced air
Buoyant force = (mass of the displaced air * acceleration due to gravity)
Since the balloon is filled with helium at standard temperature and pressure, we can approximate its density as the density of air, which is about 1.2 kg/m^3.
Now, to find the volume of the balloon, we need to determine the mass of the displaced air. We can calculate the mass by multiplying the density of air by the volume of the balloon:
Mass of the displaced air = density of air * volume of the balloon
The volume of a sphere can be calculated using the formula:
Volume of a sphere = (4/3) * π * radius^3
Now, equating the buoyant force to the total weight, we can solve for the volume:
(mass of the displaced air * acceleration due to gravity) = (total weight)
(density of air * volume of balloon * acceleration due to gravity) = (600 kg * 9.8 m/s^2) + (4000 kg * 9.8 m/s^2)
Substituting the formula for volume and solving for the radius:
(density of air * (4/3) * π * radius^3 * acceleration due to gravity) = (600 kg * 9.8 m/s^2) + (4000 kg * 9.8 m/s^2)
Now, we can solve for the radius of the balloon by rearranging the equation:
radius^3 = [(600 kg * 9.8 m/s^2) + (4000 kg * 9.8 m/s^2)] / (density of air * (4/3) * π * acceleration due to gravity)
Now, we can find the radius by taking the cube root of both sides:
radius = [ [(600 kg * 9.8 m/s^2) + (4000 kg * 9.8 m/s^2)] / (density of air * (4/3) * π * acceleration due to gravity) ]^(1/3)
I will now calculate the radius using the given values.