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 the 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 parallel movement of a straight segment along some plane closed curve. The named segment should 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 figure below.
It is not difficult to guess that this figure can be obtained by rotating the rectangle around any of its sides.
The cylinder has two identical bases, which are circles, and a lateral cylindrical 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 acquainted with the definition of a cylinder, we consider what parameters should be known in order to mathematically describe its characteristics.
The distance between the two bases is the height of the figure. Obviously, it is equal to the length of the generator. We will denote the height by 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 sufficient to uniquely describe all the properties of the figure in question.
Given the geometric definition of the cylinder, its volume can be calculated by the following formula:
V = s * h
Here S is the area of ββthe base. Note that for any cylinder and for any prism, the written formula is valid. Nevertheless, it is quite convenient to use it for a round straight cylinder, since the height is the generatrix, and the base area S can be determined by recalling the formula for the circle area:
S = pi * r 2
Thus, the working formula for volume V of the figure in question is written in the form:
V = pi * r 2 * h
Buoyancy force
Every schoolchild knows that if any object is immersed in water, then its weight will become less. The reason for this fact is the emergence of buoyant, or Archimedean force. It acts on any body, regardless of their shape and the material from which they are made. The strength of Archimedes can be determined by the formula:
F A = Ο l * g * V l
Here Ο l and V l are the density of the liquid and its volume displaced by the body. It is important not to confuse this volume with body volume. They will coincide only if the body is completely immersed in liquid. For any partial immersion, V l is always less than V of the body.
The buoyant force F A is called because it is directed vertically upward, that is, it is opposite in gravity. Different directions of the force vectors lead to the fact that the body weight in any liquid is less than in air. In fairness, we note that in the air, a buoyant force also acts on all bodies, but it is negligible compared to the Archimedean force in water (800 times less).
The difference in body weight 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 corona was made.
We use the above formula to determine the buoyancy force acting on a cylinder of brass.
The task of calculating the force of Archimedes 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 equal to the Archimedean force, which will begin to act on it if the cylinder is thrown into distilled water.
To determine the buoyancy force on a brass cylinder, first of all, in the table, see the density of brass. It is equal to 8600 kg / m 3 (this is the average value of its density). Since this value is greater than the density of water (1000 kg / m 3 ), the object will sink.
To determine the strength of Archimedes, it is enough to find the volume of the cylinder, and then use the above formula for F A. We have:
V = pi * r 2 * h = 3.14 * 5 2 * 20 = 1570 cm 3
In the formula, we substituted the value of the radius of 5 cm, since it is two times smaller than the diameter given in the condition of the problem.
For buoyancy we get:
F A = Ο l * g * V = 1000 * 9.81 * 1570 * 10 -6 = 15.4 N
Here we converted the volume V to m 3 .
Thus, an upward force of 15.4 N will act on a brass cylinder of known dimensions immersed in water.