Spatial geometry, the course of which is studied in grades 10-11 of the school, considers the properties of three-dimensional figures. The article gives a geometric definition of a cylinder, provides a formula for calculating its volume, and also solves a physical problem where it is important to know this volume.
What is a cylinder?
From the point of view of stereometry, the definition of a cylinder can be given as follows: it is a figure formed as a result of a parallel displacement of a straight segment along a certain flat closed curve. The named segment must not belong to the same plane as the curve. If the curve is a circle, and the segment is perpendicular to it, then the cylinder formed in the described way is called straight and round. It is shown in the picture below.
It's not hard to guess that this shape can be obtained by rotating a rectangle around any of its sides.
The cylinder has two identical bases, which are circles, and a sidecylindrical surface. The circle of the base is called the directrix, and the perpendicular segment connecting the circles of different bases is the generator of the figure.
How to find the volume of a round straight cylinder?
Having become familiar with the definition of a cylinder, let's look at what parameters you need to know in order to mathematically describe its characteristics.
The distance between the two bases is the height of the figure. It is obvious that it is equal to the length of the generatoratrix. We will denote the height with the Latin letter h. The radius of the circle at the base is denoted by the letter r. It is also called the radius of the cylinder. The two parameters introduced are enough to unambiguously describe all the properties of the figure in question.
Given the geometric definition of a cylinder, its volume can be calculated using the following formula:
V=Sh
Here S is the area of the base. Note that for any cylinder and for any prism, the written formula is valid. Nevertheless, for a round straight cylinder, it is quite convenient to use it, since the height is a generatrix, and the area S of the base can be determined by remembering the formula for the area of a circle:
S=pir2
Thus, the working formula for the volume V of the figure in question will be written as:
V=pir2h
Buoyancy force
Every student knows that if an object is immersed in water, then its weight will become less. The reason for this factis the emergence of a buoyant, or Archimedean force. It acts on any body, regardless of their shape and material from which they are made. The strength of Archimedes can be determined by the formula:
FA=ρlgVl
Here ρl and Vl are the density of the liquid and its volume displaced by the body. It is important not to confuse this volume with the volume of the body. They will match only if the body is completely immersed in the liquid. For any partial immersion, Vl is always less than V of the body.
The buoyant force FA is called because it is directed vertically upwards, that is, it is opposite in direction to gravity. Different directions of the force vectors lead to the fact that the weight of the body in any liquid is less than in air. In fairness, we note that in the air, all bodies are also affected by a buoyant force, however, it is negligible compared to the Archimedean force in water (800 times less).
The difference in the weight of bodies in liquid and in air is used to determine the densities of solid and liquid substances. This method is called hydrostatic weighing. According to legend, it was first used by Archimedes to determine the density of the metal from which the crown was made.
Use the above formula to determine the buoyancy force acting on a brass cylinder.
The problem of calculating the Archimedes force acting on a brass cylinder
It is known that a brass cylinder has a height of 20 cm and a diameter of 10 cm. What will be the Archimedean force,which will begin to act on him if the cylinder is thrown into distilled water.
To determine the buoyancy force on a brass cylinder, first of all, look at the density of brass in the table. It is equal to 8600 kg/m3 (this is the average value of its density). Since this value is greater than the density of water (1000 kg/m3), the object will sink.
To determine the Archimedes force, it is enough to find the volume of the cylinder, and then use the above formula for FA. We have:
V=pir2h=3, 145220=1570 cm 3
We have substituted the radius value of 5 cm into the formula, since it is two times smaller than the given one in the condition of the diameter problem.
For the buoyancy force we get:
FA=ρlgV=10009, 81157010-6 =15, 4 H
Here we have converted volume V to m3.
Thus, an upward force of 15.4 N will act on a brass cylinder of known dimensions, immersed in water.
