Conservation of angular momentum
- Mass and moment of inertia
- Flywheel
- Angular momentum
- Conservation of angular momentum
- Change in angular momentum amount
- Change in angular momentum direction
- Two ways to hand over the wheel
Mass and moment of inertia
The moment of inertia is a quantity that tells us how difficult it is to change the angular velocity of a body about a given axis of rotation. This depends on the mass of the body but also on the distribution of that mass relative to the axis of rotation.
The further the mass is from the axis of rotation, the harder it becomes to change the rotational velocity of the system (to speed it up or slow it down). Intuitively, this is because the mass now carries with it a larger amount of momentum (m·v) around the circle (due to the higher circumferential velocity) and because the momentum vector changes more rapidly. These effects depend on the distance from the axis.
In translational motion, Newton's 2nd law states that the action of a force F leads to a change in velocity (in magnitude or direction), i.e. to an acceleration a = Δv / Δt. A body "resists" any change in velocity with a property called mass (m). The greater the mass, the smaller the change in velocity of the body will be.
When a body does not translate but rotates around an axis, then the action of a torque M leads to a change in angular velocity ω, i.e. to an angular acceleration α = Δω / Δt. The body "resists" any change in angular velocity with a property called the moment of inertia (I). The greater the moment of inertia, the same torque will produce a smaller change in the angular velocity of the body.
Mass is a unique property of an object and depends only on density. Unlike mass, which is unique, an object has an infinite number of moments of inertia. Namely, the moment of inertia is tied to the axis around which we rotate the object, and the object can in principle be "pierced" in an infinite number of ways, even in such a way that the axis does not pass through the object itself at all.
Of all the possible moments of inertia, the moments of inertia for regular bodies (wheels, spheres, rods, discs, etc.) are usually calculated with respect to several characteristic axes of rotation that pass through the center of gravity of the body.
The moment of inertia is therefore a measure of the resistance to torque applied to a rotating object (i.e. the greater the moment of inertia, the slower it will accelerate when a given torque is applied, and the harder it will be to decelerate.).
Flywheel
Two wheels have transparent circular plates, so it can be seen that both are made of identical parts and therefore have the same mass. The only difference between them is the distribution of the mass in relation to the center. Let's place both bodies at the top of the slope, and let them roll simultaneously. Before we perform the experiment, we will ask a conceptual question.
– Will they roll equally or will one body be faster?
The experiment will show that the body that resists the change in rotational speed rolls faster. We conclude that wheels with a mass at a larger radius have a harder time changing angular velocity, that is, once they are already rotating, they are harder to stop. A disk with a larger moment of inertia rolls down a slope more slowly, because when converting potential energy into kinetic and rotational energy, a disk with a larger moment of inertia converts a larger part into rotational energy and a smaller part into kinetic energy. The moment of inertia allows the storage of kinetic energy of rotation. This property is used in flywheels. A flywheel is a wheel or disk that rotates, about its axis of symmetry. The energy is stored as kinetic energy, or more precisely rotational energy, and can be calculated using It is desirable for a flywheel to have most of its mass distributed on the periphery, away from the axis of rotation For translational (linear) motion, the momentum p→ is the product of the mass m and the velocity v→
p
→
= m·v
→
For rotational motion, the angular momentum L→ is the product of the moment of inertia I and the angular velocity of the object ω→
L
→
= I·ω
→
Law of conservation of momentum: The total momentum does not change. The momentum remains the same after the interaction as it was before the interaction. This law applies to both translational and angular momentum. The law of conservation of angular momentum states: if the moment of inertia I is reduced, the angular velocity ω, must be increased so that the product remains the same, i.e.
Experiments with rolling different cylinders made of copper (Cu), brass (Ms) and aluminum (Al) down a slope show that the mass, diameter (⍉) and the length of the roller (L) have no effect on the rolling speed, only the hollow roller (tube) will always lag behind the solid roller: 
Angular momentumConservation of angular momentum
Prandtl's stool
The main feature of this device is that it can rotate almost without friction around a vertical axis because it has a shaft with built-in ball bearings. The bearings should be washed with gasoline from time to time and lubricated with fine machine oil. The fact that there is no friction during rotation allows experiments to be performed in a closed system without external forces (of course, if our feet or any other part of the body do not touch the floor or surrounding objects). And this is a condition that must be met in order to be able to demonstrate the validity of the law of conservation of angular momentum, i.e. that the sum of angular momentum in such a system is constant regardless of whether the angular momentum changes as a vector, only in magnitude or only in direction.
Experiments with changing angular momentum amount
The man on the stool holds weights in his hands to increase the mass of his arms. An assistant from outside spins the man on the stool, while he holds the weights in his outstretched arms, so that the moment of inertia with respect to the axis of rotation is as large as possible. If the man now brings his arms close to his chest, the weights move closer to the axis of rotation, and therefore the moment of inertia decreases. Since, according to the law of conservation of momentum, the angular momentum must remain the same, the angular velocity increases and the rotation of the man on the stool noticeably accelerates.
The effect of the pirouette
The pirouette effect is the increase or decrease in rotational speed that occurs when the mass of a spinning body is pulled closer or further away from the axis of rotation. This is used in figure skating. Skaters first spin slowly with their arms outstretched or one leg out to the side. Pulling their arms in close to their bodies reduces the moment of inertia I. Since the law of conservation of momentum applies, the angular velocity ω→ increases and the skater spins faster. Conversely, when they extend their arms, the rotational speed decreases.
Angular momentum can be expressed as the product of the radius of gyration, mass, and angular velocity:
L→ = m·r²·ω
→
Due to the conservation of angular momentum, for a closed system without external influences and unchanged mass, where indices 1 and 2 denote the two states of the system, the following holds:
It follows from this that the angular velocities are inversely proportional to the squares of the radii of revolution of the masses:
The kinetic energy of rotation, however is not conserved because work is done against the centrifugal force while pulling the weight to a smaller radius of rotation. Let's prove this statement.
Kinetic energy
As we mentioned earlier, the conservation law states: if the moment of inertia I decreases, the angular velocity ω, must increase, i.e:
Let us assume that the kinetic energy of rotation is conserved, i.e. that the following holds true:
Which is contrary to the experiment.
We have thus proven that when pulling the weight to a smaller radius, work was obviously done against centrifugal force at the expense of the muscles of the experimenter.
Prandtl's stool without using weights
A low Prandtl stool with armrests allows the experience of conservation of angular momentum. The performer sits on the circular plate of the chair, slightly reclined and with his legs extended (position a). Someone spins him in this position. During the spin, the performer upright the body and bends his legs (position b). The speed of the spin increases rapidly. If he stretches again, the speed decreases.
2nd Experiment :We can clearly show why a cat always lands on its feet even though it is in a closed system and has no external torque to turn it. At the beginning of the experiment, a man is at rest on a stool. Then, with his outstretched arm, in which he holds a weight, he swings it above his head in circular motions around a vertical axis, during which time his body rotates in the opposite direction. As soon as the swinging of the arm stops, the rotation of the body also stops, but the body has turned. (YouTube VIDEO) When a cat falls, it wags its tail and hind legs, thus turning. At any given moment, however, the sum of the angular momentum is unchanged.
3rd Experiment :A man on a chair, in order to increase his moment of inertia, holds weights in his outstretched arms. While he is spinning, he suddenly drops the weights to the floor without changing the position of his arms in relation to the axis of rotation. The rotation of the man on the chair will remain unchanged after dropping the weights, i.e. the angular velocity will not increase, although at first glance one would think that this must happen due to the reduced moment of inertia.
A similar thing would happen if a person in free fall dropped the weights he was holding in his hands. After that, both the person and the weights would continue to fall with the same acceleration as when he was holding the weights firmly in his hand. Namely, the dropped weights took their angular momentum with them, which does not affect the rest of the system.
Experiments with changing angular momentum direction
A bicycle wheel has a mass fixed around its circumference, which gives it a large moment of inertia. The axle is extended on one side in the form of a handle, so that it can be held in the hand, and it also has a hook from which it can be hung. The ball bearings in the hub should be lubricated periodically with machine oil because it is important that the wheel has as little friction as possible. The wheel is used in experiments on the conservation of angular momentum when this quantity changes direction.
1st Experiment :
In addition to the wheel, a Prandtl's stool is also needed. While sitting on the stool, a person can be set in motion by holding the wheel axle parallel to the stool axle, with one hand and turning the wheel with the other. His body will then be pushed away from the wheel and will rotate in the opposite direction. In this case, one rotation occurs at the expense of the other, so that the sum of the angular momentum with respect to the stool axle always remains zero, as it was at the beginning. If the axle of the rotating wheel is turned 180°, then the sign of the angular momentum changes, so the rotation of the person on the stool must also change to the opposite.
When the person is on the stool (i.e. in a closed system), the rotation can only be brought about from the outside (by pushing himself off the floor or by someone standing on the floor spinning him). However, an assistant standing on the floor of the room can set the wheel in rotation and, holding the wheel axle upright, hand over the wheel to the person on the stool. The latter turns the axle 180° and thereby sets himself in rotation, so that the angular momentum is equal to twice the amount of wheel momentum that the assistant handed to him from the outside.
3rd Experiment :
If we hang a wheel by a hook at the end of its handle, it will hang with its axle in a vertical position. But if we give it a good spin before hanging it, and then hang it while it is spinning and let it go, its axle will not fall but will remain horizontal the whole time. But it will not be at rest. It will move in precession around the thread on which it is hanging. As the rotation gradually slows down due to friction, the axle will also fall, tilting from horizontal to vertical.
4th Experiment :
A wheel can be hung so that when at rest (that is, when it is not spinning) its axle is in a horizontal position. However, this requires weights to balance it. We hang weights m0 and m_, from the end of the handle, so that the wheel hangs with the axle in a horizontal position. If we spin it while it is balanced in this way and then let it go, it will not precess. But if we remove the weight m_ and only m0 remains, precession will begin in one direction (depending on the direction of rotation). When we put the removed weight back in, equilibrium is established and precession stops instantly without slowing down! If we now add to the weights m0 i m_ also a weight m+, precession will occur again, but in the other direction.
Two ways to hand over the wheel
