On a perfect day, you are hovering at low altitude in a hot-air balloon, accelerated neither upward or downward. The total wieght of the balloon, including its load and the hot air in it is 20,000N. Show that the volume of the displaced air is 1700 m3 (Hint:Buoyancy in a Gas)
You need to know the density of outside air to do this. Assume 1 atmosphere and 20 C and see what you get. The density under those conditions is about 1.19 kg/m^3, but it also depends somewhat upon the humidity
The weight of the displaced air is
(1.19) g V = 20,000 N
g is the acceleration of gravity, 9.8 m/s^2.
V = ?
The answer will not be exactly 1700 m^3, but is pretty close. Since they did not specify the air temperature, an exact answer cannot be expected.
To show that the volume of the displaced air is 1700 m³, we can use the principle of buoyancy in a gas.
According to Archimedes' principle, the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
In this case, the hot air balloon is filled with hot air, which is less dense than the surrounding air. The buoyant force acting on the balloon is equal to the weight of the air it displaces.
The weight of the balloon, including its load and the hot air in it, is given as 20,000 N.
Let's assume the density of the surrounding air is ρ₁ and the density of the hot air inside the balloon is ρ₂. The volume of the displaced air is V.
The weight of the displaced air is equal to the weight of the balloon. Hence:
ρ₁ * g * V = 20,000 N,
where g is the acceleration due to gravity.
In order to solve for V, we need to express the density of the air in terms of its volume.
The density of air can be given by the ideal gas law, which states:
ρ = (P * M) / (R * T),
where ρ is the density, P is the pressure, M is the molar mass of the gas, R is the ideal gas constant, and T is the temperature in Kelvin.
Assuming the pressure is constant and the balloon is at a constant temperature T, we can write:
ρ₁ = (P₁ * M) / (R * T) (density of the surrounding air),
ρ₂ = (P₂ * M) / (R * T) (density of the hot air inside the balloon).
Substituting these expressions into the equation for the buoyant force, we have:
(P₁ * M) / (R * T) * g * V = 20,000 N.
Rearranging this equation, we get:
V = (20,000 N * R * T) / (P₁ * M * g).
Given the values for the gas constant R (8.314 J/(mol·K)), the temperature T, the molar mass M, the pressure P₁ (assuming standard pressure at sea level), and the acceleration due to gravity g, we can calculate the volume V.
Please provide the specific values for T, M, P₁, and g, and I will calculate the volume of the displaced air, V, for you.
To determine the volume of the displaced air in the hot-air balloon, we can use the principle of buoyancy in a gas.
The buoyant force acting on the hot-air balloon is equal to the weight of the air it displaces. When the balloon is in equilibrium and not accelerating upward or downward, the buoyant force and the weight of the balloon (including its load and the hot air in it) are equal.
We are given that the total weight of the balloon, including the load and the hot air, is 20,000 N.
The buoyant force can be calculated using the equation:
Buoyant force = Density of the fluid x Volume of the displaced fluid x Acceleration due to gravity
In this case, the fluid is air, and we need to find the volume of the displaced air.
Let's assume the density of air is ρ. The weight of the balloon is equal to the total weight of the load and the hot air, which is 20,000 N. The weight of the load can be neglected since it accounts for a very small portion compared to the weight of the hot air.
Now, we can equate the buoyant force to the weight of the balloon:
Buoyant force = Weight of the balloon
ρ * Volume * g = 20,000 N
ρ * Volume = 20,000 N / g
Volume = (20,000 N / g) / ρ
To find the volume, we need to determine the density of air. Air density varies with temperature and pressure. At standard temperature and pressure (STP), the density of air is approximately 1.225 kg/m^3.
Now, we can substitute the density of air and the acceleration due to gravity to calculate the volume of the displaced air:
Volume = (20,000 N / (9.81 m/s^2)) / 1.225 kg/m^3
Simplifying the equation:
Volume = 1700 m^3
Therefore, the volume of the displaced air in the hot-air balloon is 1700 m^3.