I REALLY NEED HELP WITH THIS!!!!!
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.
Boy have you lost your mind! Because I'll help you find it!
To answer these questions, we need to consider the weight of the lead balloon, the volume of air displaced by the balloon, the buoyant force on the balloon, and whether the balloon can lift a certain payload. Let's go through each question step by step:
(a) What is the weight of the lead balloon, including helium, lead, basket, and rigging?
To find the weight, you need to determine the total mass of the lead balloon, including all the components. The mass can be calculated by adding the masses of each individual component.
Given information:
- Diameter of the balloon: 20.0 m
- Thickness of the lead foil: 0.25 mm
- Mass of the passenger basket and rigging: 295 kg
- Density of lead: 11,300 kg/m^3
- Density of helium: 0.179 kg/m^3
We need to calculate the mass of the lead foil, the mass of the helium, and then sum them up with the mass of the basket and rigging.
To calculate the mass of the lead foil:
1. Calculate the volume of the lead foil used in the sphere:
- The thickness of the lead foil is given as 0.25 mm, so the thickness of one side is 0.125 mm = 0.125 x 10^-3 m
- The diameter of the sphere is given as 20.0 m, so the radius is 10.0 m
- The volume of the lead foil can be calculated using the formula for the volume of a spherical shell: V = 4/3 * π * [(r + t)^3 - r^3], where t is the thickness of the lead foil (0.125 x 10^-3 m).
- Substitute the values into the formula and calculate the volume.
2. Calculate the mass of the lead foil using the density of lead (11,300 kg/m^3):
- Mass = density * volume
To calculate the mass of the helium:
1. Calculate the volume of the sphere (internal volume):
- The diameter of the sphere is given as 20.0 m, so the radius is 10.0 m.
- The volume of a sphere can be calculated using the formula: V = 4/3 * π * r^3.
2. Calculate the mass of the helium using the density of helium (0.179 kg/m^3):
- Mass = density * volume
Finally, to find the weight of the lead balloon, including helium, lead, basket, and rigging, add the mass of the lead foil, the mass of the helium, and the mass of the basket and rigging. Multiply the total mass by the acceleration due to gravity (9.8 m/s^2) to get the weight.
(b) What is the volume of the air displaced by the balloon? (Ignore any air displaced by the basket and rigging.)
To calculate the volume of air displaced by the balloon, use the volume of the sphere formula:
1. Calculate the volume of the sphere:
- The diameter of the sphere is given as 20.0 m, so the radius is 10.0 m.
- The volume of a sphere can be calculated using the formula: V = 4/3 * π * r^3.
(c) What is the buoyant force on the balloon?
The buoyant force on the balloon can be calculated using the Archimedes' principle:
1. Calculate the weight of the displaced air:
- The mass of the displaced air can be calculated using the density of air (1.28 kg/m^3) and the volume of air displaced by the balloon.
- Mass = density * volume
2. Calculate the buoyant force:
- Buoyant force = weight of the displaced air = mass of the displaced air * acceleration due to gravity
(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?
To determine if the balloon can lift a payload, compare the net upward force (buoyant force) on the unloaded balloon to the weight of the payload (7.5×103 N). If the net upward force is greater than or equal to the weight of the payload, then the balloon can lift it; otherwise, it cannot.
Let me know if you would like me to calculate the answers to the specific questions once you have provided the necessary calculations.