It is customary to describe bodies performing circular motions in physics using formulas that include angular velocity and angular acceleration, as well as quantities such as moments of rotation, forces, and inertia. Let us consider these concepts in more detail in the article.
The moment of rotation about the axis
This physical quantity is also called the angular momentum. The word "moment" means that when determining the appropriate characteristic, the position of the axis of rotation is taken into account. So, the angular momentum of a particle of mass m, which rotates at a speed v around the axis O and is located at a distance r from the latter, is described by the following formula:
L¯ = r¯ * m * v¯ = r¯ * p¯, where p¯ is the particle momentum.
The sign "¯" indicates the vector character of the corresponding quantity. The direction of the vector of the moment of rotation L¯ is determined by the rule of the right hand (four fingers are directed from the end of the vector r¯ to the end of p¯, and the laid-off thumb indicates where L¯ will be directed). The directions of all these vectors can be seen on the main photo of the article.
When solving practical problems, they use the formula for the angular momentum in scalar form. In addition, the linear speed is replaced by the angular. In this case, the formula for L will look like this:
L = m * r 2 * ω, where ω = v * r is the angular velocity.
The value m * r 2 is denoted by the letter I and is called the moment of inertia. It characterizes the inertial properties of the rotation system. In general terms, the expression for L is written as follows:
L = I * ω.
This formula is valid not only for a rotating particle of mass m, but also for any body of arbitrary shape that performs circular displacements about some axis.
Moment of inertia I
In the general case, the value I introduced in the previous paragraph is calculated by the formula:
I = ∑ i (m i * r i 2 ).
Here, i indicates the number of an element with mass m i located at a distance r i from the axis of rotation. This expression allows us to calculate for an inhomogeneous body of arbitrary shape. For most ideal volumetric geometric figures, this calculation has already been performed, and the obtained values of the moment of inertia are listed in the corresponding table. For example, for a homogeneous disk that performs circular motions around an axis perpendicular to its plane and passing through the center of mass, I = m * r 2/2.
To understand the physical meaning of the moment of inertia of rotation I, one must answer the question about which axis is easier to spin the squeegee: one that runs along the squeegee or one that is perpendicular to it? In the second case, you will have to make more efforts, since the moment of inertia for this position of the mop is large.
The law of conservation of L
The change in torque in time is described by the formula below:
dL / dt = M, where M = r * F.
Here M is the moment of the resulting external force F applied to the shoulder r relative to the axis of rotation.
The formula shows if M = 0, then a change in the angular momentum L will not occur, that is, it will remain unchanged for an indefinitely long time regardless of internal changes in the system. This case is written as an expression:
I 1 * ω 1 = I 2 * ω 2 .
That is, any changes within the system of moment I will lead to changes in the angular velocity ω in such a way that their product will remain constant.
An example of the manifestation of this law is an athlete in figure skating, who, throwing his hands and pressing them to the body, changes his I, which is reflected in a change in his rotation speed ω.
The task of the Earth's rotation around the Sun
We will solve one interesting problem: using the above formulas, it is necessary to calculate the moment of rotation of our planet in its orbit.
Since the attraction of other planets can be neglected, and also taking into account that the moment of gravitational force acting from the side of the Sun to the Earth is equal to zero (shoulder r = 0), then L = const. To calculate L, we use the following expressions:
L = I * ω; I = m * r 2 ; ω = 2 * pi / T.
Here we accepted that the Earth can be considered a material point with a mass m = 5.972 * 10 24 kg, since its size is much less than the distance to the Sun r = 149.6 million km. T = 365.256 days - the period of revolution of the planet around its star (1 year). Substituting all the data in the expression above, we get:
L = I * ω = 5.972 * 10 24 * (149.6 * 10 9 ) 2 * 2 * 3.14 / (365.256 * 24 * 3600) = 2.66 * 10 40 kg * m 2 / s.
The calculated value of the angular momentum is gigantic, due to the large mass of the planet, the high speed of its rotation in orbit and the huge astronomical distance.