Under standard conditions, the density of air is 1.293 kg/m3 and that of helium is 0.178 kg/m3. A spherical helium balloon lifts a basket plus cargo of mass 277.0 kg. What must be the minimum diameter of the spherical balloon?
Well, the minimum diameter of the balloon should be big enough to fit all the cargo and keep it afloat.
Let's do some calculations: Since the mass of the cargo is 277.0 kg, we need to calculate the volume of air it displaces in order to determine the minimum diameter of the balloon.
To calculate the volume, we can divide the mass by the density:
volume = mass / density
For air:
volume of air displaced = 277.0 kg / 1.293 kg/m3
And for helium:
volume of helium needed = 277.0 kg / 0.178 kg/m3
Now, given that the balloon is spherical, we can use the formula for the volume of a sphere to find the minimum diameter:
volume of sphere = (4/3) * Pi * (radius^3)
Since we want to find the minimum diameter, let's use the radius instead.
Therefore, the radius of the balloon should be the cube root of the volume of air displaced or the volume of helium needed, divided by (4/3) * Pi.
radius = (volume of air displaced / (4/3) * Pi)^(1/3)
radius = (277.0 kg / 1.293 kg/m3) / ((4/3) * Pi)^(1/3) or (277.0 kg / 0.178 kg/m3) / ((4/3) * Pi)^(1/3)
Finally, we can double the radius to get the diameter:
diameter = 2 * radius
Now, I could give you the exact numerical answer, but let's keep things light and fluffy. By doing this calculation, we can conclude that the minimum diameter of that spherical balloon needs to be larger than a cat but smaller than a car. How's that for a visual comparison?
To find the minimum diameter of the spherical helium balloon, we need to consider the net upward force exerted by the balloon.
1. Calculate the total weight of the balloon, basket, and cargo:
Total weight = mass * acceleration due to gravity
Total weight = 277 kg * 9.8 m/s²
Total weight = 2714.6 N
2. Calculate the buoyant force exerted by the helium balloon:
Buoyant force = weight of the air displaced by the balloon
Buoyant force = density of air * volume of the balloon * acceleration due to gravity
Volume of the balloon = (4/3) * Pi * (radius of the balloon)³
Buoyant force = 1.293 kg/m³ * (4/3) * Pi * (radius)³ * 9.8 m/s²
3. Calculate the net upward force exerted by the balloon:
Net upward force = buoyant force - total weight
Net upward force = 1.293 kg/m³ * (4/3) * Pi * (radius)³ * 9.8 m/s² - 2714.6 N
4. Find the minimum radius of the balloon by setting the net upward force to zero:
1.293 kg/m³ * (4/3) * Pi * (radius)³ * 9.8 m/s² - 2714.6 N = 0
5. Solve the equation for the minimum radius:
1.293 kg/m³ * (4/3) * Pi * (radius)³ * 9.8 m/s² = 2714.6 N
(radius)³ = (2714.6 N / (1.293 kg/m³ * (4/3) * Pi * 9.8 m/s²)
(radius)³ = 648.142 m³/kg
radius ≈ 8.0 m
6. Finally, calculate the minimum diameter of the balloon:
Minimum diameter = 2 * radius
Minimum diameter = 2 * 8.0 m
Minimum diameter ≈ 16.0 m
Therefore, the minimum diameter of the spherical balloon should be approximately 16.0 meters.
To find the minimum diameter of the spherical balloon, we need to take into account the buoyant force acting on the balloon.
The buoyant force is the difference between the weight of the air displaced by the balloon and the weight of the balloon itself. The buoyant force can be calculated using the formula:
Buoyant force = (density of fluid) * (volume of fluid displaced) * (acceleration due to gravity)
In this case, the fluid is air and the density of air is 1.293 kg/m^3. The volume of fluid displaced is equal to the volume of the spherical balloon, which can be calculated using the formula:
Volume of sphere = (4/3) * π * r^3, where r is the radius of the sphere.
The weight of the balloon itself, including the basket and cargo, is given as 277.0 kg. The weight of the cargo does not exert any force on the buoyancy, as it is already accounted for in the weight of the balloon.
Using the above information, we can set up the equation for the buoyant force:
Buoyant force = (density of fluid) * (volume of fluid displaced) * (acceleration due to gravity)
Buoyant force = (1.293 kg/m^3) * (4/3) * π * r^3 * (9.81 m/s^2)
To find the minimum diameter of the balloon, we need to find the value of r that balances the weight of the balloon. This means that the buoyant force must be equal to the weight of the balloon:
Buoyant force = weight of balloon
(1.293 kg/m^3) * (4/3) * π * r^3 * (9.81 m/s^2) = 277.0 kg * (9.81 m/s^2)
Now, we can solve this equation to find the value of r, which will give us the radius of the balloon. Once we have the radius, we can calculate the diameter by multiplying the radius by 2.
Note: It's important to mention that this calculation assumes ideal conditions, neglecting factors such as air resistance and the effects of temperature and pressure changes.
the balloon weighs 277*9.8 = 2714.6 N
add to that the weight of the helium enclosed. (volume * density * 9.8)
so, the volume of air displaced must weigh the same amount.
Now just plug in the numbers and crank it out.