Many problems of motion in classical mechanics can be solved using the concept of momentum of a particle or the entire mechanical system. Let us consider in more detail the concept of impulse, and also show how the acquired knowledge can be used to solve physical problems.
The main characteristic of movement
In the 17th century, when studying the movement of celestial bodies in space (the rotation of planets in our solar system), Isaac Newton used the concept of momentum. In fairness, we note that several decades earlier, a similar characteristic was already used by Galileo Galilei in describing bodies in motion. However, only Newton was able to concisely integrate it into the classical theory of the movement of celestial bodies that he developed.
Everyone knows that one of the important quantities characterizing the speed of a change in the coordinates of a body in space is speed. If we multiply it by the mass of a moving object, then we will get the mentioned amount of motion, that is, the following formula is true:
p¯ = m * v¯
As can be seen, p¯ is a vector quantity whose direction coincides with that for the speed v¯. It is measured in kg * m / s.
The physical meaning of p¯ can be understood in the following simple example: a truck rides at the same speed and a fly flies, it is clear that a person cannot stop the truck, but a fly can without problems. That is, the amount of movement is directly proportional not only to speed, but also to body weight (depends on inertial properties).
The motion of a material point or particle
When considering many problems of motion, the size and shape of a moving object often do not play a significant role in solving them. In this case, one of the most common approximations is introduced - the body is considered a particle or a material point. It is an immense object, the whole mass of which is concentrated in the center of the body. This convenient approximation is true when the dimensions of the body are much smaller than the distances traveled by it. A striking example is the movement of a car between cities, the rotation of our planet in its orbit.
Thus, the state of the particle in question is characterized by the mass and speed of its movement (note that the speed may depend on time, that is, not be constant).
What is a particle impulse?
Often these words mean the momentum of a material point, that is, p¯. This is not entirely correct. We will examine this issue in more detail, for this we write the second law of Isaac Newton, which is already in the 7th grade of the school, we have:
F¯ = m * a¯
Knowing that acceleration is the rate of change of v¯ in time, we rewrite it as follows:
F¯ = m * dv¯ / dt => F¯ * dt = m * dv¯
If the acting force does not change with time, then for the interval Δt the equality is true:
F¯ * Δt = m * Δv¯ = Δp¯
The left side of this equality (F¯ * Δt) is called the momentum of force, the right side (Δp¯) is the change in momentum. Since the case of the motion of a material point is considered, this expression can be called the particle momentum formula. It shows how much its total momentum will change in time Δt under the action of the corresponding force impulse.
Momentum
Having dealt with the concept of momentum of a particle of mass m for linear motion, we proceed to consider a similar characteristic for circular motion. If a material point, having a momentum p¯, rotates around the O axis at a distance r¯ from it, then we can write this expression:
L¯ = r¯ * p¯
This expression represents the angular momentum of the particle, which, like p¯, is a vector quantity (L¯ is directed according to the rule of the right hand perpendicular to the plane constructed on the segments r¯ and p¯).
If the momentum p¯ characterizes the intensity of the linear movement of the body, then L¯ has a similar physical meaning only for a circular trajectory (rotation around the axis).
The formula for the angular momentum of the particle, written above, in this form is not used to solve problems. Using simple mathematical transformations, we can arrive at the following expression:
L¯ = I * ω¯
Where ω¯ is the angular velocity, I is the moment of inertia. This notation is similar to that for a linear particle momentum (analogy between ω¯ and v¯ and between I and m).
Conservation laws of p¯ and L¯
In the third paragraph of the article, the concept of an impulse of external force was introduced. If such forces do not act on the system (it is closed, and only internal forces take place in it), then the total momentum of the particles belonging to the system remains constant, that is:
p¯ = const
Note that as a result of internal interactions, each coordinate of the momentum is conserved:
p x = const .; p y = const .; p z = const
Usually this law is used to solve problems with the collision of solids, such as balls. It is important to know that no matter what character the collision has (absolutely elastic or plastic), the total momentum will always remain the same before and after the impact.
Drawing a complete analogy with the linear motion of a point, we write the conservation law for the angular momentum as follows:
L¯ = const. or I 1 * ω 1 ¯ = I 2 * ω 2 ¯
That is, any internal changes in the moment of inertia of the system lead to a proportional change in the angular velocity of its rotation.
Perhaps one of the common phenomena that demonstrate this law is the rotation of the skater on ice, when he groups his body differently, while changing his angular velocity.
The task of colliding two sticky balls
Let us consider an example of solving the problem of preserving the linear momentum of particles moving towards each other. Let these particles be balls having a sticky surface (in this case, the ball can be considered a material point, since its size does not affect the solution of the problem posed). So, one ball moves along the positive direction of the X axis at a speed of 5 m / s, it has a mass of 3 kg. The second ball moves along the negative direction of the X axis, its speed and mass are 2 m / s and 5 kg, respectively. It is necessary to determine in which direction and at what speed the system will move after the collision of the balls and their adhesion to each other.
The momentum of the system before the collision is determined by the difference in the momentum for each ball (the difference is taken because the bodies are directed in different directions). After the collision, the momentum p¯ is expressed by only one particle whose mass is m 1 + m 2 . Since the balls only move along the X axis, we have the expression:
m 1 * v 1 - m 2 * v 2 = (m 1 + m 2 ) * u
From where the unknown speed is found by the formula:
u = (m 1 * v 1 - m 2 * v 2 ) / (m 1 + m 2 )
Substituting the data from the condition, we get the answer: u = 0.625 m / s. A positive value of speed indicates that the system after the impact will move in the direction of the X axis, and not against it.