In this problem, your task is to analyze a lead balloon. The balloon in question is a hollow sphere of diameter of 20.0 m, constructed from a lead foil 0.25 mm thick and filled with helium gas. Below the balloon, a passenger basket is suspended. The mass of the passenger basket along with the balloon's riggings is 295 kg. For various parts of this problem, you will need to know the density of lead (11,300 kg/m3), and the densities of air and helium at standard atmospheric temperature and pressure (1.28 kg/m3 and 0.179 kg/m3). (a) What is the weight of the lead balloon, including helium, lead, basket and rigging? (b) What is the volume of the air displaced by the balloon? (Ignore any air displaced by the basket and rigging.) (c) What is the buoyant force on the balloon? (d) The maximum "payload" of the balloon, the weight of the passengers and equipment it can lift, equals the net upward force on an unloaded balloon. Can this balloon lift a payload of 7.5×103 N?
Volume of sphere =π•D³/6 = 4188.8 m³,
Surface of sphere = π•D² = 1256.6 m²,
Volume of lead foil = 1256.6•0.00025 = 0.314 m³
Mass of the lead foil = ρ•V= 11300•0.314 = 3548.2 kg.
(a) Weight = mg=[m(sph)+m(passenger)+m(He)] •g=
= [3548.2+295+749.8] •9.8=45011.4 N.
(b) 4188.8 m³,
(c) F= ρ•V•g=1.28•4188.8•9.8 = 52544.3 N.
(d) Δ= 52544.3-45011.4=7532.9 N
7532.9>7500 => This balloon can lift the payload.
To answer the questions, let's go through each step:
(a) To calculate the weight of the lead balloon, including helium, lead, basket, and rigging, we need to find the total mass first.
The mass of the lead foil can be found by calculating the volume of the lead and multiplying it by the density of lead. The volume of the lead can be found by subtracting the volume of the empty space in the hollow sphere from the volume of the entire sphere.
The volume of the sphere is given by V = (4/3)πr^3, where r is the radius of the sphere (half of the diameter).
In this case, the diameter is 20.0 m, so the radius is 10.0 m. Plugging it into the formula, we get V = (4/3)π(10.0)^3 = (4/3)π(1000) ≈ 4188.79 m^3.
To find the volume of the hollow space, we need to subtract the volume of the inner sphere with a radius equal to the radius of the lead foil (10.0 m - 0.25 mm).
The volume of the inner sphere is V_inner = (4/3)π(9.99975)^3 ≈ 4181.62 m^3.
Therefore, the volume of the lead in the foil is V_lead = V - V_inner ≈ 4188.79 m^3 - 4181.62 m^3 ≈ 7.17 m^3.
Now, we multiply the volume of the lead by the density of lead to find its mass: Mass_lead = V_lead * Density_lead = 7.17 m^3 * 11,300 kg/m^3 ≈ 80,941 kg.
The mass of the balloon, including helium, lead, basket, and rigging, is the sum of the individual masses: Mass_balloon = Mass_lead + Mass_helium + Mass_basket.
Given that the mass of the basket and rigging is 295 kg, we assume that the helium gas fills the entire hollow space, so its mass can be found by multiplying the volume of the hollow space by the density of helium: Mass_helium = V_lead * Density_helium = 7.17 m^3 * 0.179 kg/m^3 ≈ 1.28 kg.
Finally, the weight of the balloon is given by Weight_balloon = Mass_balloon * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
(b) To find the volume of the air displaced by the balloon, we need to determine the volume of the balloon itself, as it displaces an equal volume of air.
We already calculated the volume of the entire sphere in (a) as 4188.79 m^3.
Therefore, the volume of the air displaced by the balloon is also 4188.79 m^3.
(c) The buoyant force on the balloon is the weight of the volume of air displaced by the balloon.
Buoyant force = Weight of displaced air = Density_air * Volume_displaced_air * g.
Given that the density of air is 1.28 kg/m^3, and the volume of the displaced air is 4188.79 m^3 (calculated in (b)), the buoyant force can be found using the formula:
Buoyant force = 1.28 kg/m^3 * 4188.79 m^3 * 9.8 m/s^2.
(d) The maximum "payload" of the balloon is the net upward force on an unloaded balloon. To check if the balloon can lift a payload of 7.5×10^3 N, we need to compare it with the buoyant force.
If the buoyant force is greater than or equal to the payload weight, the balloon can lift the payload.
Therefore, we compare the buoyant force from (c) to the payload weight of 7.5×10^3 N to determine if the balloon can lift it.