A cylinder is one of the simple three-dimensional figures that is studied in the school geometry course (section solid geometry). In this case, problems often arise to calculate the volume and mass of the cylinder, as well as to determine its surface area. Answers to the marked questions are given in this article.
What is a cylinder?
Before proceeding to the answer to the question, what is the mass of the cylinder and its volume, it is worth considering what this spatial figure is. It should be noted right away that a cylinder is a three-dimensional object. That is, in space, you can measure three of its parameters along each of the axes in a Cartesian rectangular coordinate system. In fact, to unambiguously determine the dimensions of a cylinder, it is enough to know only two of its parameters.
Cylinder is a three-dimensional figure formed by two circles and a cylindrical surface. To more clearly represent this object, it is enough to take a rectangle and start rotating it around any of its sides, which will be the axis of rotation. In this case, the rotating rectangle will describe the shaperotation - cylinder.
Two round surfaces are called the bases of the cylinder, they are characterized by a certain radius. The distance between the bases is called the height. The two bases are interconnected by a cylindrical surface. The line passing through the centers of both circles is called the axis of the cylinder.
Volume and surface area
As you can see from the above, the cylinder is defined by two parameters: the height h and the radius of its base r. Knowing these parameters, it is possible to calculate all other characteristics of the considered body. Below are the main ones:
- The area of the bases. This value is calculated by the formula: S1=2pir2, where pi is the number of pi equal to 3, 14. Digit 2 in formula appears because the cylinder has two identical bases.
- Cylindrical surface area. It can be calculated like this: S2=2pirh. It is easy to understand this formula: if a cylindrical surface is cut vertically from one base to another and expanded, then a rectangle will be obtained, the height of which will be equal to the height of the cylinder, and the width will correspond to the circumference of the base of the three-dimensional figure. Since the area of the resulting rectangle is the product of its sides, which are equal to h and 2pir, the above formula is obtained.
- Cylinder surface area. It is equal to the sum of the areas of S1 and S2, we get: S3=S 1 + S2=2pir2 + 2pirh=2pi r(r+h).
- Volume. This value is easy to find, you just need to multiply the area of one base by the height of the figure: V=(S1/2)h=pir2 h.
Determining the mass of a cylinder
Finally, it's worth going directly to the topic of the article. How to determine the mass of a cylinder? To do this, you need to know its volume, the formula for calculating which was presented above. And the density of the substance of which it consists. The mass is determined by a simple formula: m=ρV, where ρ is the density of the material that forms the object in question.
The concept of density characterizes the mass of a substance that is in a unit volume of space. For example. It is known that iron has a higher density than wood. This means that in the case of equal volumes of iron and wood matter, the former will have a much larger mass than the latter (approximately 16 times).
Calculating the mass of a copper cylinder
Consider a simple problem. It is necessary to find the mass of a cylinder made of copper. For definiteness, let the cylinder have a diameter of 20 cm and a height of 10 cm.
Before you start solving the problem, you should deal with the source data. The radius of the cylinder is equal to half of its diameter, which means r=20/2=10 cm, while the height is h=10 cm. Since the cylinder considered in the problem is made of copper, then, referring to the reference data, we write out the density value of this material: ρ=8, 96 g/cm3 (for temperature 20 °C).
Now you can start solving the problem. First, let's calculate the volume: V=pir2h=3, 14(10)210=3140 cm3. Then the mass of the cylinder will be: m=ρV=8.963140=28134 grams or approximately 28 kilograms.
You should pay attention to the dimension of the units during their use in the corresponding formulas. So, in the problem, all parameters were presented in centimeters and grams.
Homogeneous and hollow cylinders
From the result obtained above, it can be seen that a copper cylinder with relatively small dimensions (10 cm) has a large mass (28 kg). This is due not only to the fact that it is made of heavy material, but also to the fact that it is homogeneous. This fact is important to understand, because the above formula for calculating the mass can only be used if the cylinder is completely (outside and inside) made of the same material, that is, it is homogeneous.
In practice, hollow cylinders are often used (for example, cylindrical barrels for water). That is, they are made of thin sheets of some material, but inside they are empty. For a hollow cylinder, the indicated formula for calculating the mass cannot be used.
Calculating the mass of a hollow cylinder
It is interesting to calculate what mass a copper cylinder will have if it is empty inside. For example, let it be made from a thin copper sheet with a thickness of only d=2 mm.
To solve this problem, you need to find the volume of the copper itself, from which the object is made. Not the volume of the cylinder. Because the thicknessthe sheet is small compared to the dimensions of the cylinder (d=2 mm and r=10 cm), then the volume of copper from which the object is made can be found by multiplying the entire surface area of the cylinder by the thickness of the copper sheet, we get: V=dS 3=d2pir(r+h). Substituting the data from the previous problem, we get: V=0.223, 1410(10+10)=251.2 cm3. The mass of a hollow cylinder can be obtained by multiplying the obtained volume of copper, which was required for its manufacture, by the density of copper: m \u003d 251.28.96 \u003d 2251 g or 2.3 kg. That is, the considered hollow cylinder weighs 12 (28, 1/2, 3) times less than a homogeneous one.
