Background:When flying in a lighter-than-air balloon, the load you carry cannot weigh more than what the balloon can carry. Even many years ago, scientists knew how much weight a balloon could lift because of Archimedes Principle.
Four measurements are used to calculate the total weight that a balloon can lift:
Say that you are an aeronaut who had planned to take your three cousins on a balloon ride to see the countryside from the air. You also planned to bring a picnic lunch to feed everyone (including yourself!) and,since it is cooler up in the air,you wanted to bring some blankets to keep everyone warm. Just when you had everything ready to go, your cousin from far away paid a surprise visit and wanted to go along also. Given all the facts below, can your cousin go along?
In preparation for the flight, you had calculated the total weight of all the people and equipment you expected to bring along. Your calculations were as follows:
|You (the aeronaut)||
|weight of helium gas||
|ropes and other equipment||
|lunch for four people||
|blankets for four people||
Your balloon,with brilliant red,white and blue stripes, can carry 22,500 cubic feet of gas. You can say that the volume of your balloon is 22,500 cubic feet. You also know from your study of chemistry that the density of helium is .011 pounds per cubic foot. If you multiply the volume of the balloon, 22,500 cubic feet, by the density of helium, 0.011 pounds per cubic feet, you get the weight of the helium.
22,500 cubic feet X 0.011 pounds per cubic feet = 248 pounds of helium
To calculate how many pounds your balloon could carry, you multiplied the density of the air by the volume of the balloon. Notice that the weight of the air displaced equals the total weight the balloon can lift. As Archimedes said: " The buoyant force on a body is equal to the weight of the gas (or fluid) displaced by the body."
Density of the air=0.0755 pounds per cubic foot
Volume of the balloon =22,500 cubic feet
0.0755 pounds per cubic foot X 22,500 cubic feet =1700 pounds
So, based on your calculations, you could carry 1700 pounds on your flight .On the list above, where you totaled the weight of all the items you expected to carry, you expected to carry 1566 pounds.
Would you be able to fly your three cousins, plus lunch and blankets on your balloon? Yes, because they weighed 1566 pounds and you could carry 1700!
But what about your cousin Bryant who wanted to come along? Bryant tells you that he weighs 160 pounds. You add Bryant's weight to the total weight of all the items you expected to carry:
1566 pounds + 160 pounds =1726 pounds
Oh, no! Bryant cannot fly with you! Can you tell why?
That's right! The reason is that with the addition of Bryant the total weight of the items you want to carry becomes too big for the balloon you have.
What can you do so Bryant can go along? Well, you calculated that your balloon could carry 1700 pounds. How much over that limit are you if Bryant comes along?
1726 pounds - 1700 pounds =26 pounds
So, you must remove 26 pounds from your weight list. Obviously you cannot remove the people, the gondola, the balloon or the ropes and other equipment. What's left?
The total weight of the lunch and blankets is:
20 pounds + 8 pounds =28 pounds
So, if you left the lunch and blankets at home, the total weight you need to carry is:
1726 pounds - 28 pounds =1698 pounds
Without the lunch and blankets the total weight of 1698 pounds is less than the 1700-pound limit that your balloon can carry. You need to make a decision! Bryant may come along on the flight and the lunch and blankets stay home, or the lunch and blankets come along and Bryant stays home. What will you do?
Use the template below to help you complete the exercises.
Step 1: Fill in the following weight list table. List the items you want to take along in the left-hand column,and the weight of each item in the right-hand column.
Step 2: Calculate the total weight your balloon can carry. Assume that you will fill your balloon with helium gas.
Density of the air =0.0755 pounds per cubic foot
Volume of the balloon =_____________________cubic feet
0.0755 pounds per cubic foot X ___________cubic feet =________pounds the balloon can carry
Step 3:Determine whether the balloon can carry all of the items you want to take with you.
Is the total weight of all the items you want to carry greater than, less than, or equal to the weight the balloon can carry? Circle the correct answer.
is less than _________________________pounds is greater than _____________________pounds total weight is equal to weight the balloon can carry
Will the balloon carry all of the items you want to take?________(yes/no)
Step 4: If your answer was "yes," then Bon Voyage!
If your answer was "no," then you need to go back to your Weight Table and figure out what can be left at home, then recalculate your Weight Table and Step 3.
What are some other steps you could take to lighten the load?
Question for Extended Thinking:
What is the importance of including the weight of the gas in the total weight of the balloon?
What if the gas is hot air?
Hint: The weight of hot air inside the balloon is not very different than the weight of the air displaced by the balloon. There is a relationship between the density of air and the temperature of air. You can tell what the difference in density is for the hot air compared to the cold air outside.