Density problems are typical and uncomplicated, they are offered to solve for students in grade 7. The article provides the main nuances that must be considered when solving problems of this type. Visual examples with solutions are also provided.
Mass and volume
Before considering the problems of density physics, it is necessary to define some quantities that will be discussed. First of all, it is mass and volume.
Mass is an integral characteristic of a body, which reflects how "heavy" it is. This value is directly related to the amount of substance, that is, the more atoms in the composition the body in question contains, the greater its mass. This means that we are talking about an additive physical characteristic. The mass is usually denoted by the letter m or M. In SI, it is expressed in kilograms (kg), but its other units (tons, grams, milligrams) are often used.
Many people confuse this value with body weight. Indeed, in ordinary life we ββare used to talking about the weight of objects, and not about their mass. Both quantities are related to each other through a constant: P = m * g, where P is the weight, and g = 9.81 m / s 2 is the acceleration that the earth's gravity forces to any object during its fall in airless space.
Volume is the geometric characteristic of any bodies, which reflects what part of the space they occupy. Since our space is three-dimensional, and each of the dimensions has a unit of length, the volume is expressed in these units, cubed. In SI it will be cubic meters (m 3 ), but often use km 3 , cm 3 . Volumes of liquid and loose bodies are often represented in liters. Volume, as a rule, in physical formulas is denoted by the letter V (note that this is the capital letter, since small v is often denoted by speed). This quantity, like mass, is additive. For example, if 1 liter of water is added to 1 liter of water, then its total volume will double.
Body density
As noted above, every body has a certain mass m and occupies a specific volume V. From a mathematical point of view, density is the coefficient of proportionality between the noted values. From the point of view of physics, density indicates how much mass (weight) a given body of unit volume will have.
Density is an important characteristic of a substance, that is, if different bodies are made of the same material, then their density will be the same. It is usually denoted by the Greek letter Ο (po).
According to the above definition, we can write the following equality:
Ο = m / V.
It can be seen that the unit Ο is kilogram per cubic meter (kg / m 3 ). Nevertheless, other units may be used, for example, g / cm 3 . The ratio of two additive quantities (mass and volume) leads to a nonadditive, i.e. density does not depend on the geometric dimensions of the body, nor on its weight.
Next, we give examples of problems on density, mass and volume in order to better absorb the knowledge gained. To solve these problems, you must have at hand a table with densities for different substances.
Archaeologists and elephant figurine
Consider the solution to the following interesting problem on body density: during archaeological excavations, a small figurine of an elephant was found. To verify what it is made of, scientists first measured its volume by immersing an object in water, and then mass, weighing it on a scale. What result did archaeologists get if it is known that the volume of the figurine was 42 cm 3 and its mass was 810 grams?
Since both quantities, m and V, are known, we can immediately substitute them in the formula given in the previous paragraph of the article. Then we get: Ο = 810/42 = 19.286 g / cm 3 . Attention should be paid to the dimensions of the initial values, which led to the corresponding units of the calculated density. Now, if we turn to the density table, we can see that gold has a value of Ο = 19.32 g / cm 3 . This number is very close to what we got. Therefore, we can conclude: the elephant figurine is made of gold (since the number 19.286 is slightly less than 19.32, there are some light impurities in the gold found by archaeologists).
Which cylinder has the highest density?
This question must be answered in order to solve the following density problem, in which two cylinders are considered. The mass and volume of the first of them are equal to 0.5 kg and 900 cm 3 , and for the second these figures are 1.2 kg and 2160 cm 3 .
Before you substitute known data into the formula, you should agree on their dimensions. In particular, it is necessary to convert the original mass units to grams. Then, 0.5 kg = 500 g and 1.2 kg = 1200 g. Now we substitute these values ββin the expression for density, we get for the 1st cylinder: Ο 1 = 500/900 = 0.556 g / cm 3 . For the second: Ο 2 = 1200/2160 = 0.556 g / cm 3 . It follows that Ο 1 = Ο 2 , that is, the densities of the cylinders are the same. In addition, by this figure we can say that the cylinders are made of wood.
Sand truck
In a series of solutions to density problems, we consider another interesting example. It is known that the truck has a body, the linear dimensions of which are 4x2x1.5 meters. This machine is capable of carrying loads of up to 15 tons. Can a full body of sand be poured for transport?
First, we calculate the volume of the body. This is done by simply multiplying its linear parameters, that is, V = 4 * 2 * 1.5 = 12 m 3 . Now we calculate the maximum density of the material that a truck can transport if it fills its entire body. We have: Ο = 15000/12 = 1250 kg / m 3 (15 tons are equal to 15000 kg).
If you look at the tabular data for the density of sand, you can see that it is about 1500 kg / m 3 . Since this value is greater than the obtained figure, the body of this truck cannot be completely filled with sand.
What is the density of our planet?
We will solve this curious problem of body density. To do this, you first need to find the volume of the Earth, knowing that its average radius is R = 6371 km. This can be done by the formula: V = 4/3 * pi * R 3 , where pi = 3.1416. We get: V = 1,0832 * 10 21 m 3 . The mass of the Earth can be found from the literature, it is 5.972 * 10 24 kg. Where do we get the average value for the density of the planet: Ο = 5.972 / 1.0832 * 1000 = 5513 kg / m 3 .