Cylinders and Cones
Ever wondered why all rockets have similar shapes?
In the past 60 years, rocket technology has advanced significantly, and more than 35 000 rockets have been sent to space.
While launching a rocket to the space, there are three main forces you need to be aware of:
Gravity, air resistance and thrust
The shape of the rockets designed to minimize the air resistance called drag. Smooth, round surfaces produce less friction so cause
Therefore, it is clear that we need
Rockets are cylindrical in body and conical at the top.
We can think of cylinders as circular versions of prisms. Similarly, we say that the circular version of a pyramid is a cone.
Use the slider to increase the number of sides of the pyramid.
As the number of sides increases, the prism starts to look more like a cylinder.
Use the slider to increase the number of sides of the prism.
As the number of sides increases, the pyramid starts to look more like a cone.
Cylinders consist of two congruent, parallel circles joined by a curved surface. These circles are the bases** of the cylinder.
The cylindrical part of the rockets accommodates the rocket's essential components like liquid oxygen, hydrogen tanks and engines.
In fact all the pressure vessels like fuel tankers are round, since round shapes provide maximum strength from internal pressure. A cylindrical shape adds less weight of the rocket’s walls. Cylinders also don’t have any “weak points” like the edges of the
The only problem with cylindrical structures is the drag rate. This value can be reduced slightly by adding a cone to the top.
The conical shape can be in different forms according to the purpose of the rocket.
A cone has
Nose cones of the rockets usually carry payloads like satellite, cargo or passengers. If it is the external fuel tank, it carries the liquid oxygen tank.
Nose cones are also designed for all the aircrafts like planes and zeppelins as well as the underwater and in high-speed land vehicles.
While the shape of the rockets are mostly similar, they can be in different sizes according to the type of the mission.
What is the height of a rocket with an orbital mission carrying 23 tons payload?
Type of the mission affects the amount of fuel that the rocket has to carry. The significant portion of the rockets’ volume is held by fuel tanks, therefore the size of the rocket mostly depends on the fuel tanks.
For an orbital mission, fuel tanks must have the capacity to hold approximately 550 thousands liters of liquid oxygen and 1.5 million liters of hydrogen.
In a space shuttle, the external fuel tank carries a cylindrical liquid hydrogen tank.
The volume of the cylinder is a measurement describing how much (in cubic units) the cylinder will hold. It is a measure of the space inside the cylinder.
Think of a stack of coins as a model of a cylinder. The number of coins you put on top of each other is actually the
So the space you have created with the coin stack is basically
Like the prisms, a cylinder has a volume equal to the product of its
Recall that the ratio of the area of a circle to its square radius is a constant number called
Pi is an
This indicates that the decimal part of the π goes to infinity without repeating itself.
π = 3.1415926…
The volume of a cylinder is,
{.text-center}
{.text-center}
The cylindrical hydrogen tank has a diameter of approximately 8 meters. It needs to hold 1.5 million liters of hydrogen.
Remember
So the volume of the hydrogen tank has to be around
The liquid oxygen tank is on the other hand, located at the top of the external tank and has a conical shape.
Remember that cones are the
Even though a cone is technically not a pyramid, they share many properties.
To calculate the volume of a cone, can we still use the same formula with the pyramids?
The ratio of the volumes of a pyramid and a prism with the base of same size and shape and the same height is
Let’s repeat the same experiment to see how many cones of water are needed to fill the cylinder with the same radius and height.
It takes the contents of
Nose cone has the same radius with the cylinder and has to hold 550 thousand liters of oxygen. Therefore the conical tank has to have a volume of 550 cubic meters.
which is around 30 meters.
The real shape of the cone is larger than the one that we have calculated here and has a height of almost 17 meters. So the total height of the External Tank is around 50 meters.
When full with the fuel, External tank alone weighs 760 thousands kilograms. Rocket companies try to increase the efficiency of the rockets by decreasing the overall weight and increasing the payload capacity.
In time, many attempts were made to reduce the huge amount of weight of the tanks. Standard weight tanks are evolved to super lightweight tanks by examining every little detail of the rockets carefully.
1981 Space Shuttle Columbia vs 1982 Space Shuttle Columbia
Inıtially, fuel tanks were painted white to protect them from ultraviolet light damage. Engineers then realized that did not cause a problem and wondered if no paint is used, how many kilograms of weight can be spared?
To be able to find the amount of paint used, we calculate the
Again, we can think of the cylinder and the cone separately.
Surface area of a cylinder:
We can use the
The net of the cylinder has two
The height of the rectangle is
Since the lateral face of the cylinder is a rectangle, we can find the area by
It is basically the
Therefore, the lateral area is
So the Surface Area of a cylinder is the sum of all the face areas.
The surface area of the cylindrical part of the tank with a height of 30 meters and a diameter of 8 meters . For the rocket model we need to calculate
Surface area of a cone:
Move the slider to see the net of the cone. We have a
“s” is called the slant height of the cone, the same as the pyramids. Slant height is
Now we just have to add up the area of both faces;
The base of the cone is a circle with radius r, so its area is
We may think the lateral area as one large triangle consists of infinitely many triangles.The total length of the triangle’s base is the
Recall that the area of a triangle is one
Then, the total area of the cone is
To find the amount of the paint used, we need to calculate the
The Conical part of the tank has a slant height of 20 meters and a diameter of 8 meters.
The total surface area of the rocket is the sum of the area of the cylindrical body
If a gallon of paint is used to cover a 12 square meters, to cover the entire tank, almost
Approximate weight of a gallon of exterior paint is almost 3 kilograms. Therefore, by not painting the tank, the engineers spared
So far, we have learnt the surface area and the volume calculations of the cylinders and the cones as well as the unique properties of these solids that make them right choices for the rocket design as well as the other drag reducing land and underwater vehicles..
If we consider our initial question again, we may have a look at another property of the cylinder.
Let’s compare the volumes of different prisms with the same base perimeter and height;
It also turns out that
Earlier, we have seen that cylinders have the best properties as a container whereas the cones are usually used as roofs, shelters, baskets, etc..
Throughout history, conic huts and roofs are built and used by different civilizations all over the world in different places and times. There is even a whole town in Italy, famous for its unique conic shaped trulli roofs.
Conical Hats, Asia
Native American Teepees
Alberobello, Italy
Also, there are lots of cone-shaped rock formation examples all around the world. But most probably the first thing comes to our mind when we think of the conical shapes are the roofs of the Earth; Mountains and Volcanoes
Kasha-Katuwe
rock formations, New Mexico
Chimney rocks
Cappadocia, Turkey
Everest Mt 8850 m
Earth
Olympus Mons 21 000 m
Mars
Mountains and volcanoes maintain the interesting conical shape of these rock formations but they are much bigger.
The Highest Mountain of our world, Everest, is growing taller up to 5 centimeters each year. Can it grow infinitely and become as high as the Olympus Mons of Mars? Is there a limit for the maximum possible height of a mountain on Earth?
There is an interesting fact about the height of the mountains.
By approximating the shape of the mountains to
When we solve the inequality here, we will see that a mountain on Earth can be maximum around
This is just a little bit higher than Mt Everest!
If 8800 meters-tall-Everest continues to grow around 5 cm a year, almost
But do not worry! We still have another mountain which is in fact much higher than Mt Everest.
Just it is on Mars.
The highest volcano in the solar system is Olympus Mons on Mars with a height of over 21 km and a radius of 312 km.
You can change the constants from Earth to Mars to calculate the maximum height of a mountain on Mars.
With a rough approximation since the gravity of Mars is almost
PROBLEM SOLVING
A grain silo is another example of the usage of cylinders and cones. It usually is built from two cones and a cylinder in between. Silos are used in agriculture to store grain. Find the capacity of the giant grain silo 12 by 20m?
To find the amount of grain that the silo holds, we need to calculate the
Top and Bottom cones have a radius of
The volume of each cone is:
Cylinder has the
The total volume of the grain silo is
May the Less Drag and the new heights be with you !