When solving problems on the motion of bodies in space, the formulas for the conservation of kinetic energy and momentum are often used. It turns out that similar expressions exist for rotating bodies. This article discusses in detail the law of conservation of angular momentum (the corresponding formulas are also given) and gives an example of solving the problem.
Rotation process and angular momentum
Before proceeding to consider the formula of the law of conservation of angular momentum, it is necessary to get acquainted with this physical concept. The easiest way to enter it is to use the figure below.
The figure shows that at the end of the vector r¯, directed from the axis of rotation and perpendicular to it, there is some material point of mass m. This point moves in a circle of the named radius with linear velocity v¯. It is known from physics that the product of mass and linear velocity is called the momentum (p¯). Now it’s worth introducing a new value:
L¯ = r¯ * m * v¯ = r¯ * p¯.
Here, the vector quantity L¯ represents the angular momentum. To go to the scalar form of writing, it is necessary to know the moduli of the corresponding values of r¯ and p¯, as well as the angle θ between them. The scalar formula for L has the form:
L = r * m * v * sin (θ) = r * p * sin (θ).
In the figure above, the angle θ is right, so you can simply write:
L = r * m * v = r * p.
From the recorded expressions it follows that the unit of measurement for L will be kg * m 2 / s.
Direction of the angular momentum vector
Since the quantity in question is a vector (the result of a vector product), it will have a certain direction. From the properties of the product of two vectors, it follows that their result will give a third vector perpendicular to the plane formed by the first two. At the same time, it will be directed in such a way that if you look from its end, the body will rotate counterclockwise.
The result of applying this rule is shown in the figure in the previous paragraph. It can be seen from it that L¯ is directed upwards, since if you look at the body from above, its movement will occur against the clock hands. When solving problems, it is important to consider the direction during the transition to the scalar form of recording. So, the considered angular momentum is considered positive. If the body rotated clockwise, then in the scalar formula in front of L should put a minus sign (-L).
Linear momentum analogy
Considering the topic of the angular momentum and the law of its conservation, we can do one mathematical trick - transform the expression for L¯, multiplying and dividing it by r 2. Then it turns out:
L¯ = r * m * v¯ * r 2 / r 2 = m * r 2 * v¯ / r.
In this expression, the ratio of speed to radius of rotation is called angular velocity. It is usually denoted by the letter of the Greek alphabet ω. This value shows how many degrees (radian) the body will rotate in the orbit of its rotation per unit time. In turn, the product of mass per square radius is also a physical quantity that has its own name. Designate it I and call the moment of inertia.
As a result, the formula for the angular momentum is converted into the following form of writing:
L¯ = I * ω¯, where ω¯ = v¯ / r and I = m * r 2 .
The expression demonstrates that the direction of the angular momentum L¯ and the angular velocity ω¯ coincide for a system consisting of a rotating material point. Of particular interest is the value I. Below it is considered in more detail.
Moment of inertia of the body
The introduced value I characterizes the "resistance" of the body to any change in the speed of its rotation. That is, it plays exactly the same role as the inertia of the body during linear movement of the object. In fact, I for a circular motion from a physical point of view means the same as mass in a linear motion.
As it was shown, for a material point with mass m rotating about an axis r at a distance from it, the moment of inertia is easy to calculate (I = m * r 2 ), however, for any other bodies this calculation will be somewhat complicated, since it involves the use of the integral.
For a body of arbitrary shape, I can be determined using the following expression:
I = ∫ m (r 2 * dm) = ∫ V (r 2 * ρ * dV), where ρ is the density of the material.
The expressions above mean that to calculate the moment of inertia, one should divide the whole body into infinitesimal volumes dV, multiply them by the square of the distance to the axis of rotation and by density and add up.
For bodies of various shapes, this problem is solved. The following are data for some of them.
Material point: I = m * r 2 .
Disc or cylinder: I = 1/2 * m * r 2 .
A rod of length l, fixed in the center: I = 1/12 * m * l 2 .
Ball: I = 2/5 * m * r 2 .
The moment of inertia depends on the distributed body mass relative to the axis of rotation: the farther away from the axis the greater part of the mass will be, the more I will be for the system.
The change in angular momentum in time
Considering the angular momentum and the law of conservation of angular momentum in physics, one can solve a simple problem: determine how and due to which it will change in time. To do this, take the derivative with respect to dt:
dL¯ / dt = d (r¯ * m * v¯) / dt = m * v¯ * dr¯ / dt + r * m * dv¯ / dt.
The first term here is equal to zero, since dr¯ / dt = v¯ and the product of vectors v¯ * v¯ = 0 (sin (0) = 0). The second term can be rewritten as follows:
dL¯ / dt = r * m * a¯, where the acceleration a = dv¯ / dt, whence:
dL¯ / dt = r * F¯ = M¯.
The quantity M¯, by definition, is called the moment of force. It characterizes the action of the force F¯ on a material point of mass m located at a distance r from the axis of rotation.
What does the resulting expression show? It demonstrates that a change in the angular momentum L¯ is possible only due to the action of the angular momentum M¯. This formula is the law of conservation of the angular momentum of a point: if M¯ = 0, then dL¯ / dt = 0 and L¯ is a constant.
What moments of forces can change L¯ systems?
There are two types of moments of forces M¯: external and internal. The former are associated with the force acting on the elements of the system by any external forces, while the latter arise due to the interaction of parts of the system.
According to Newton’s third law, any force of action corresponds to the opposite directed reaction force. This means that the total moment of force of any interactions within the system is always zero, that is, it cannot affect changes in the angular momentum.
The value of L¯ can change only due to external moments of forces.
Formula of the law of conservation of angular momentum
The formula for writing the conservation expression for the quantity L¯ if the sum of the external moments of forces is zero has the following form:
I 1 * ω 1 = I 2 * ω 2 .
Any changes in the moment of inertia of the system are proportionally reflected in the change in the angular velocity so that the product I * ω does not change its value.
The form of this expression is similar to the law of conservation of linear momentum (the role of mass is played by I, and the role of speed is ω). If we develop the analogy further, then, in addition to this expression, we can write another one that will reflect the conservation of kinetic energy of rotation:
E = I * (ω) 2/2 = const.
Application of the law of conservation of angular momentum finds itself in a number of processes and phenomena, which are briefly described below.
Examples of using the law of conservation of L¯
The following examples of the law of conservation of angular momentum are important for the respective fields of activity.
- Any sport where you need to make jumps with rotation. For example, a ballerina or figure skating athlete begins to perform a pirouette with rotation, spreading her arms wide and moving her leg away from the center of gravity of her body. Then he presses the leg closer to the supporting one and the arm closer to the body, thereby reducing the moment of inertia (most of the points of the body are located close to the axis of rotation). According to the law of conservation of L, its angular velocity of rotation ω should increase.
- To change the orientation direction relative to the Earth of any artificial satellite. This is done like this: the satellite has a special heavy "flywheel", it is driven by an electric motor. The total angular momentum must be maintained, so the satellite itself begins to rotate in the opposite direction. When it takes the desired orientation in space, the flywheel is stopped, and the satellite also stops rotating.
- The evolution of stars. As the star burns its hydrogen fuel, the forces of gravity begin to prevail over the pressure of its plasma. This fact leads to a decrease in the radius of the star to small sizes and, as a consequence, to a strong increase in the angular rotation speed. For example, it was found that neutron stars having a diameter of several kilometers rotate at gigantic speeds, making one revolution in fractions of a millisecond.
Solution of the problem of the conservation law L¯
Scientists have found that after several billion years the Sun, having exhausted energy reserves, will turn into a "white dwarf". It is necessary to calculate how fast it will rotate around its axis.
First you need to write out the values of the necessary quantities, which can be taken from the literature. So, now this star has a radius of 696,000 km and makes one revolution around its axis in 25.4 Earth days (value for the equator region). When it comes to the end of its evolutionary path, it will shrink to the size of 7000 km (on the order of the radius of the Earth).
Assuming that the Sun is an ideal ball, one can use the formula of the law of conservation of angular momentum to solve this problem. You need to translate the day into seconds and kilometers to meters, it turns out:
L = I * ω = 2/5 * m * r 1 2 * ω 1 = 2/5 * m * r 2 2 * ω 2 .
From where it follows:
ω 2 = (r 1 / r 2 ) 2 * ω 1 = (696000000/7000000) 2 * 2 * 3.1416 / (25.4 * 24 * 3600) = 0.0283 rad / s.
Here we used the formula for the angular velocity (ω = 2 * pi / T, where T is the rotation period in seconds). When performing the calculations, it was also assumed that the mass of the Sun remains constant (this is not true, since it will decrease. Nevertheless, the obtained value of ω 2 is the lower boundary, that is, in reality, the dwarf Sun will rotate even faster).
Since a full revolution is 2 * pi radians, then it turns out:
T 2 = 2 * pi / ω 2 = 222 s.
That is, at the end of its life cycle, this star will make one revolution around its axis faster than in 222 seconds.