Many problems in physics can be successfully solved if the laws of conservation of one or another quantity are known during the physical process under consideration. In this article, we consider the question of what is the momentum of the body. And the law of conservation of momentum will also be carefully studied.
General concept
More correctly, we are talking about the amount of movement. Galileo first began to study the patterns associated with it at the beginning of the 17th century. Based on his writings, Newton during this period published a scientific work. In it, he clearly and clearly stated the basic laws of classical mechanics. Both scientists by the amount of movement understood the characteristic, which is expressed by the following equality:
p = m * v.
Based on it, the value p determines both the inertial properties of a body of mass m and its kinetic energy, which depends on the speed v.
Momentum is called momentum because its change is related to the momentum of force through Newton’s second law. Show it is not difficult. It is only necessary to find the derivative of the momentum in time:
dp / dt = m * dv / dt = m * a = F.
Where do we get:
dp = F * dt.
The right side of the equation is called the impulse of power. It shows the magnitude of the change in momentum over time dt.
Closed systems and internal forces
Now we should deal with two more definitions: what is a closed system, and what are internal forces. Let's consider in more detail. Since we are talking about mechanical motion, then a closed system is understood to mean a set of objects that external bodies do not affect in any way. That is, in such a structure, the total energy and the total amount of matter are preserved.
The concept of internal forces is closely related to the concept of a closed system. By such are meant only those interactions that are realized exclusively between the objects of the structure under consideration. That is, the action of external forces is completely excluded. In the case of movement of the bodies of the system, the main types of interaction are mechanical collisions between them.
Determination of the law of conservation of body momentum
The momentum p in a closed system, in which only internal forces act, remains constant for an arbitrarily long time. It cannot be changed by any internal interactions between bodies. Since this quantity (p) is vectorial, this statement should be applied to each of its three components. The formula for the law of conservation of momentum of the body can be written as follows:
p x = const;
p y = const;
p z = const.
This law is conveniently applied in solving practical problems in physics. In this case, the one-dimensional or two-dimensional case of the motion of bodies before their collision is often considered. It is such a mechanical interaction that leads to a change in the momentum of each body, but their total momentum remains constant.
As you know, mechanical collisions can be absolutely inelastic and, conversely, elastic. In all these cases, the momentum is conserved, although in the interactions of the first kind, the kinetic energy of the system is lost as a result of its conversion to heat.
Task example
Having become acquainted with the definitions of the momentum of the body and the law of conservation of momentum, we will solve the following problem.
It is known that two balls, each with a mass of m = 0.4 kg, roll in the same direction with speeds of 1 m / s and 2 m / s, while the second follows the first. After the second ball caught up with the first one, an absolutely inelastic collision of the bodies under consideration occurred, as a result of which they began to move as a whole. It is necessary to determine the joint speed of their translational motion.
To solve this problem is not difficult if you apply the following formula:
m * v 1 + m * v 2 = (m + m) * u.
Here, the left side of the equality represents the momentum before the collision of the balls, the right - after the collision. The speed u will be equal to:
u = (m * v 1 + m * v 2 ) / (2 * m) = (v 1 + v 2 ) / 2;
u = 1.5 m / s.
As you can see, the final result does not depend on the mass of the balls, since it is the same.
Note that if, according to the condition of the problem, the collision would be absolutely elastic, then to obtain the answer, one should use not only the conservation law of p, but also the conservation law of the kinetic energy of the ball system.
Body rotation and angular momentum
All that was said above relates to the translational movement of objects. The dynamics of rotational motion is in many ways similar to its dynamics with the difference that it uses the concepts of moments, for example, moment of inertia, angular momentum and angular momentum. The latter is also called angular momentum. This value is determined by the following formula:
L = p * r = m * v * r.
This equality means that in order to find the angular momentum of a material point, we must multiply its linear momentum p by the radius of rotation r.
After an angular momentum, Newton’s second law for the motion of rotation is written in the following form:
dL = M * dt.
Here M is the moment of force that acts on the system during the time dt, giving it angular acceleration.
The law of conservation of angular momentum of the body
The last formula in the previous paragraph of the article suggests that a change in the value of L is possible only if some external forces act on the system, creating a non-zero moment of rotation M. In the absence of such, the value of L remains unchanged. The law of conservation of angular momentum states that no internal interactions and changes in the system can lead to a change in the module L.
If we use the concepts of inertia of moment I and angular velocity ω, then the conservation law under consideration can be written in the form:
L = I * ω = const.
It manifests itself when, during the performance of a number with rotation in figure skating, the athlete changes the shape of his body (for example, presses his hands to the body), while his moment of inertia changes and is inversely proportional to the angular velocity.
Also, this law is used to make turns around the own axis of artificial satellites during their movement in orbit in outer space. In the article, we examined the concept of the momentum of a body and the law of conservation of momentum of a system of bodies.