During the video clips Prof. kstra posed the question: "How many helium party balloons could you safely carry before getting airborne?" This is exactly what we will find out now!
Suppose that a party balloon is spherical, has a radius of 16 centimetres and is filled with helium. Furthermore suppose that we do our experiment under standard atmospheric conditions, when the air density is 1.225 . How many of these balloons should a 75 kilogram human carry in order to become airborne?
4178
How do you find this number?
Well, if we're talking about becoming airborne like a superhero, then I'm afraid party balloons won't quite do the trick. You might end up looking more like a floating parade float than a crime-fighting superhero.
But let's do the math for fun anyway! The first thing we need to calculate is the volume of one balloon. The formula for the volume of a sphere is (4/3)πr^3, where r is the radius.
So, for a balloon with a radius of 16 centimeters, the volume would be (4/3)π(16^3).
Once we have the volume, we can calculate the weight of the displaced air using the density of air. The formula for weight is density times volume, so let's multiply the volume by the density of air, which is 1.225.
Now we have the weight of the displaced air by one balloon. To figure out how many balloons a 75-kilogram human would need to carry to become airborne, we divide 75 kilograms by the weight of the displaced air by one balloon.
And... after all that math, I'm sorry to say that you would need a lot more than just a few party balloons to become airborne. It's more like a bazillion balloons, which might make you look more like a funny floating clown than a superhero. But hey, at least it would make for a great party decoration!
To find out how many helium party balloons a 75 kilogram human should carry in order to become airborne, we need to calculate the lift force exerted by the balloons and compare it to the weight of the human.
Step 1: Calculate the volume of one balloon:
The volume of a spherical balloon can be calculated using the formula:
V = (4/3) * π * r^3
where r is the radius of the balloon.
Using the given radius of 16 centimeters:
V = (4/3) * π * (16 cm)^3
V = (4/3) * π * (4096 cm^3)
V ≈ 21,493.34 cm^3
Step 2: Convert the volume to cubic meters:
1 cubic meter = 1,000,000 cubic centimeters
Therefore, the volume of one balloon in cubic meters is approximately 0.02149334 m^3.
Step 3: Calculate the buoyant force exerted by one balloon:
Buoyant force (Fb) = Volume of displaced fluid * Density of fluid * Gravity
In this case, the fluid being displaced is air and the density of air under standard conditions is 1.225 kg/m^3.
Fb = 0.02149334 m^3 * 1.225 kg/m^3 * 9.8 m/s^2
Fb ≈ 0.2578 kg * m/s^2
Step 4: Calculate the total lift force exerted by multiple balloons:
Assuming each balloon provides the same lift force, we can multiply the lift force of one balloon by the number of balloons.
Let's assume x is the number of balloons needed to become airborne.
Total lift force (Flift) = x * Fb
Step 5: Compare the lift force with the weight of the human:
The weight of the human is given as 75 kilograms.
If the total lift force is greater than or equal to the weight of the human, the human will become airborne.
Flift ≥ 75 kg * 9.8 m/s^2
Solving for x:
x * Fb ≥ 75 kg * 9.8 m/s^2
x ≥ (75 kg * 9.8 m/s^2) / Fb
Plugging in the values:
x ≥ (75 kg * 9.8 m/s^2) / 0.2578 kg * m/s^2
Calculating:
x ≥ 289.93
Therefore, a 75 kilogram human would need to carry approximately 290 helium party balloons in order to become airborne.
To determine the number of helium party balloons a 75-kilogram human would need to carry in order to become airborne, we'll need to consider a few factors.
First, let's calculate the volume of a single helium balloon. Since the balloon is spherical, we can use the formula for the volume of a sphere:
V = (4/3)πr³
Where:
V is the volume of the balloon
π is a mathematical constant approximately equal to 3.14159
r is the radius of the balloon
Given that the radius of the balloon is 16 centimeters (0.16 meters), we can substitute this value into the equation:
V = (4/3)π(0.16)³
V = (4/3)π(0.016)
Next, let's calculate the buoyant force acting on a single balloon. The buoyant force is the force exerted by a fluid (in this case, air) on an object immersed in it. It opposes the force of gravity.
The buoyant force can be calculated using Archimedes' principle:
Buoyant Force = (Density of Air) * (Volume of Displaced Air) * (Acceleration Due to Gravity)
The density of air under standard atmospheric conditions is given as 1.225 kilograms/m³.
To calculate the volume of displaced air, we'll need to know the density of helium. Helium is less dense than air, so it will displace some of the air when encapsulated in a balloon.
The density of helium is approximately 0.179 kilograms/m³.
Now, let's calculate the volume of displaced air when a single helium balloon is released. Since the balloon is filled with helium, its volume will be equal to the volume of the balloon.
The volume of displaced air can be calculated as follows:
Volume of Displaced Air = Volume of Balloon = V
Substituting the value of V calculated earlier, we have:
Volume of Displaced Air = (4/3)π(0.016)
Now we can substitute the values into the formula for the buoyant force:
Buoyant Force = (Density of Air) * (Volume of Displaced Air) * (Acceleration Due to Gravity)
Buoyant Force = (1.225) * (V) * (9.8)
Now we have the buoyant force acting on a single balloon.
Finally, to determine the minimum number of balloons required to become airborne, we need to overcome the gravitational force acting on a 75-kilogram human.
Gravitational Force = Mass * Acceleration Due to Gravity
Gravitational Force = (75) * (9.8)
To become airborne, the buoyant force acting on the balloons must be greater than the gravitational force acting on the human:
Number of Balloons = Gravitational Force / Buoyant Force
Number of Balloons = (75 * 9.8) / ((1.225) * (V) * (9.8))
Now, you can substitute the value of V calculated earlier into the equation and solve for the number of balloons required to become airborne.