If you want to be able to argue planetary motion, or electronic motion, or even if you want to defend the roundness of the Earth, you're going to need to master the concept that we're going to introduce in this video. I'm talking about Angular Momentum. It's the rotational analogue of linear momentum that we saw here on the channel earlier.
So, get ready, because the content you're going to see now is a great show! Hey everyone, welcome to Scientific Verve. I'm Eudes Fileti, professor and scientist and here on this channel, I present the most important concepts of science for those who want to learn a little more about the principles that govern the world.
One of these principles we are going to see in this video. If you like this idea, subscribe to this channel, click this button! Also activate your bell so you can receive notifications whenever we launch something new here.
ANGULAR MOMENT When someone starts running on a treadmill, the tape on the treadmill also starts to move, but in the opposite direction. If an animal walks on a rotating platform, the platform will rotate in the opposite direction. Drlls are used to drill the floor, wall, wood.
In drills, while the bit rotates in one direction, its motor and housing tend to rotate in the opposite direction. The helicopter's cabin rotates in the opposite direction to its main rotor. To prevent this from happening, a lateral rotor rotating in a vertical plane is placed on the tail of the helicopter.
This tail rotor counteracts the cabin's spin, pushing in the air and stabilizing the helicopter's motion. In the case of two-rotor helicopters, the principle is simpler. Here, we simply have one rotor that rotates in one direction, while the other rotates in the opposite direction.
And the result is that no cab rotation is felt. Two boys in swivel chairs lift their feet off the floor and try to push each other, causing a spin. What happens is that both end up rotating: one in the opposite direction to the other.
In all these cases, we observe that the origin of the rotation of an object is in the rotation movement of another object in the opposite direction. And this occurs in such a way that it is as if one object wanted to compensate for the movement of the other. Which suggests to us that something, some property, is being conserved.
This is where the concept of angular momentum, the rotational analogue to linear momentum, comes into play. DEFINITION OF ANGULAR MOMENTUM Angular momentum is the momentum that a rotating object acquires as it rotates. The greater the angular momentum, the more difficult it is to stop the object from rotating.
As you may have already noticed, every magnitude of rotation has an analogue in translation. So the definition for angular momentum is not going to deviate from this pattern. We saw in this video here, that the definition of linear momentum is mass times velocity.
So, by analogy, it is accepted that the definition of angular momentum is moment of inertia times angular velocity. Here, the moment of inertia is the analogue of mass, while the angular velocity is the analogue of linear velocity. The unit of measurement for angular momentum in the international system is the kilogram meter squared per second.
DEFINITION OF ANGULAR MOMENTUM The most general case is defined by the vector product of the distance to the axis of rotation and the linear moment. That is, the angular momentum is R x P or R x mv. This implies that angular momentum is always perpendicular to the object's orbit.
Note that the orbit is defined by the position vector and the velocity vector. Equivalently, we can say that the angular momentum will define the plane of the orbit. So if you change the direction of angular momentum, you change the plane of the orbit accordingly.
So by this definition, we see that the greater the moment of inertia or the angular velocity, the greater the angular momentum. These two ways of defining angular momentum can be illustrated by Earth motions. In rotation, the Earth rotates around its own axis.
Although it has a relatively low angular velocity, its moment of inertia due to its large mass distribution is enormous. With that it presents a huge angular momentum related to its rotation. This is the angular momentum of rotation, or spin momentum.
In addition to rotation, we know that the Earth has its orbital translation movement, approximately circular, along its orbit. So due to this movement the Earth has another moment, the orbital angular momentum. This moment is much greater than the moment of gyration and can be obtained using the definition in terms of its mass, radius, and linear velocity.
RELEVANCE OF ANGULAR MOMENT As with energy and linear momentum, angular momentum is a fundamental property and is observed at all size scales. In particular, angular momentum is fundamental in quantum mechanics, because in addition to being the basic unit of Planck's constant, which is the characteristic constant of the quantum world, it is also a key concept for describing the behavior of the atom and its cloud. electronics.
There are important quantum mechanics books dedicated exclusively to describing the angular momentum of atomic and subatomic systems. CONSERVATION OF ANGULAR MOMENT Just as linear momentum is conserved in the absence of external forces, so is angular momentum when there is no net external torque. When you push a carousel, turn a bicycle wheel, or open a door, you exert torque.
Rotation speeds up and angular momentum increases. So, in a given time interval, you observe a change in angular momentum. The greater the resulting torque, the faster this ΔL variation will be.
The relationship between torque and the change in angular momentum is given by the expression on the side. Note that this expression is exactly analogous to the relationship between force and linear momentum that we obtained earlier. It is essential in mechanics as it is a way of expressing Newton's 2nd of rotation.
Through it, we can understand several relatively complex phenomena, such as why the Earth does not stop rotating. We can write this expression as follows: ∑τ = ΔL/Δt. This means that in order to change angular momentum, a torque must act over some period of time.
What's more, it means that if the resulting torque is zero, the angular momentum will be constant or conserved. In this case, we must have ΔL = 0, so that the moment of a system at the beginning of the event must be equal to the moment at the end of the event. So, if its angular velocity increases after the event, necessarily the angular momentum after the event must have been reduced, and vice versa.
EXAMPLES OF CONSERVATION OF ANGULAR MOMENTUM Before the mouse began its movement, there was no rotation in the system. So the total angular momentum of the system before is zero. When the mouse starts to run the treadmill, it rotates in the opposite direction to the movement of the mouse.
So, during the movement, the angular momentum of the mouse compensates the angular momentum of the treadmill. So, the total angular momentum will remain zero. The faster the mouse runs, the faster the treadmill will rotate in the opposite direction.
If there were two identical mice on the treadmill running at the same speed, the treadmill would rotate twice as fast as if there was only one mouse. Now, if the rats were allowed to run in opposite directions, no treadmill rotation could be observed. This is the same as when two boys try to push each other in the chair.
One rotates in one direction and the other rotates in the opposite direction. Prior to spin, the angular momentum is zero because there is no rotation. Now, during the spin, we see that the momentum of one is opposite the angular momentum of the other.
This leads to zero total angular momentum. The moment is preserved! Here it is interesting to note that the bigger boy will rotate more slowly than the smaller boy.
This is because the inertia of one is greater than the inertia of the other. This is compatible with conservation of angular momentum. So whoever has the greatest inertia will necessarily have the smallest angular velocity.
An example of conservation of angular momentum is given by an ice skater performing a spin. The resulting torque on it will be virtually zero, as there is little friction between your skates and the ice. Consequently, it can spin for some time, and its angular momentum will be conserved during the spin.
As the skater spins, she can increase her angular velocity simply by pulling in her arms and legs. And so it increases its rotation speed. When she does this, retracting her arms and legs, the skater's mass redistributes closer to her axis of rotation, and this reduces her moment of inertia.
Therefore, the angular velocity must increase to keep the angular momentum constant. So does a person in a swivel chair. By closing her arms holding dumbbells, she drastically decreases her moment of inertia and, as a result, starts to spin at a much higher speed.
When you open your arms, moving the dumbbells away, your speed decreases again. AN INTERESTING DIFFERENCE And this is an interesting difference between the conservation principles of angular momentum and linear momentum. In the linear case, a body will never increase or decrease its speed just by the action of internal forces.
For this, it is necessary to apply a force or have friction so that the body changes its state of motion. This was seen in this video here. Now this is not true for angular momentum.
A rotating body can change its angular velocity simply through internal forces, such as muscular effort, which alters the moment of inertia around the axis. CENTRAL FORCES In the movement of a satellite, the moon for example, the force of attraction is directed towards the center of the Earth. Forces like this, which have a line of action through the two interacting bodies, are called central forces.
Movements caused by central forces will always have their angular momentum conserved. Stressing: movements due to central forces will always have conserved angular momentum. This is important.
Since the moon is attracted to Earth by a central force, its angular momentum around Earth will remain constant. Therefore, the torque of the gravitational force must be zero. Remember, torque is proportional to the change in angular momentum.
If the variation is zero, the angular momentum is constant. And the torque is zero. For the same reason, Earth's orbital angular momentum around the Sun is conserved.
Incidentally, this is what keeps Earth orbiting, as there is no external torque capable of reducing its angular momentum. And hence its angular velocity. The same goes for the angular momentum of rotation.
The only torques exerted are due to the gravitational interactions of the Sun and Moon, as the Earth is not a perfect sphere. But the effects of these torques are extremely small and affect Earth's angular velocity very little. A RELEVANT QUESTION You might still wonder why, when using a drill, the bit rotates one way, but the drill housing does not rotate the other way.
Even if the person feels the carcass tend to turn to the other side, it does not. So, does this mean that the angular momentum, in this case, was not conserved? No, what happens here is that the system in question it does not just consist of the bit and the drill housing.
Whoever holds the drill is firmly supported by the Earth. But, as the moment of inertia "drill + person + planet" is much greater than that of the drill, the speed acquired by this huge system is negligible. And what we see in practice is that it is null.
The same thing happens when a person spins an object attached to a string. The inertia of the person-planet system is so great that its reaction spin cannot be perceived. Finally, we can make a parallel to all the facts discussed here in the channel between translational movement, linear; and angular rotational motion, here in this frame.
Every linear property has a corresponding angular analogue. With that, we close the discussion on rotational motion. We are able to discuss some of its most notable applications, including the theory of gravitation.
But that will be a topic for future videos. Click this button here in the middle of the screen to subscribe to the channel. And have Scientific Verve.
Hugs and see you next time.
