Lec33 物理(一) Newton’s 2nd Law in Uniform Circular Motion — Transcript

motion and other applications, so we're going to talk about Newton's mechanics, but we're going to talk about circular motion or other more special forces and movements.

Okay, we'll divide this part into three sections. The first part we're going to introduce is Newton's Second Law in Uniform Circular Motion, which everyone is familiar with, uniform circular motion.

Okay, let's look at Section 1, Newton's Second Law in Uniform Circular Motion. Actually, we have already learned Newton's First Law, and we also talked about the Second and Third Laws earlier.

Now, the First Law tells you that when the net force is zero, an object at rest stays at rest, and an object in motion stays in motion with constant velocity in a straight line. The Second Law tells you about inertia, which is mass, so it's about the relationship between force, mass, and acceleration.

So, if you really follow these rules of physics to think about everything around you, because our world is a physical world, from the smallest things around you, even smaller to what we call semiconductor technology, which has reached 5 nanometers, 3 nanometers, you have to be able to think about it.

And if you do, you've already learned that the first law tells you that you can only go straight, right? Now, the question is, why doesn't it go straight? Why does it curve?

And if it curves, you'll find that the speed will always be tangent to the curve, which you'll also learn in vector calculus. So, you'll find that at the next time point, its speed will still be tangent to the curve and will turn. Now, what we're going to look at is a common occurrence in daily life.

That is, we tie a stone with a string and start spinning it around. When you spin it around, if there's no specific force, it will spin at a constant speed, meaning the magnitude of the speed is the same, and it will make a curve or circular motion. So, now we need to think about this problem.

That is, I encounter it in daily life, and this is my physical world. This problem mostly has the same speed magnitude, meaning the speed is the same, so you can write that the magnitude is still V. But its direction has changed. So, here we have to see what is the reason for its turn.

So, it should have a mechanism behind it. You can't use the first law anymore. The first law tells you that it should go straight. The second law says that you are subjected to a force. And earlier, we said that force only causes acceleration. But here, it tells you that there is another acceleration.

That acceleration is called centripetal acceleration. Its concept is that you must first be subjected to this centripetal force to have centripetal acceleration, and then your trajectory can turn. So, the concept comes down to this: your original speed is in this direction, let's say it's in this direction.

And then later, you are forced to turn, so it becomes parallel to the tangent of the tangent point. So, you should have a force pulling inward. This force pulling inward, you will eventually find that when you learn vector calculus, it is related to the curvature of the curve. And curvature is what? Curvature is that any non-straight line will have a curvature, kappa.

If the radius of the circle is R, then it is 1/R. So, all the stories are integrated together. And then you will find that this force pulling inward, which we call centripetal force, you must first have centripetal force to turn. This centripetal force should also be related to the curvature. The answer is yes, you have derived it before.

The answer is that it is mass times R over V squared. And 1/R is the curvature, kappa. So, all the relationships are integrated together. So, let's look at this. The so-called centripetal force should first have centripetal force, and then generate centripetal acceleration. Centripetal acceleration. Then it can make your movement state take the next step and turn, so there will be this curved movement.

So, you must first have this centripetal force. Let's look at some statements. It says, the centripetal force is given at first, then the object can turn its moving direction. This is a descriptive statement. It says that there is first centripetal force, and then the object can turn. The second point is that the centripetal force is always directed to the center of the trajectory. Find out the plane of the circular trajectory at first. Actually, this is a very interesting phenomenon.

That is, you first have centripetal force, and then you point to the center. Then you cause that curved motion. But when you analyze this curved motion, you also find that you can calculate the curvature. The curvature is the degree of its bending. This curvature is actually also pointing to the center. So, this is a chicken-and-egg situation. That is, if you don't have centripetal force, you won't turn. And if you turn, you will have this curvature. This curvature is also pointing to the center of the circle, a radius. The curvature is inversely proportional to the radius. The larger the radius, the smaller it is. So, the whole concept of chicken and egg. This centripetal force is very, very important in daily life. Because we have learned friction before, and friction is always used in daily life. Centripetal force is also. Why? Because you can't always go straight. On Earth, you always go straight, and then you have to make a big turn. When you turn, you must have centripetal force. You see, for example, when you ride a bicycle, you lean when you turn. When you lean, it creates centripetal force, allowing you to turn. So, if you don't lean, you can't turn. When you drive, it's the same. All the elevated roads are curved, and the road surface is also inclined. They are all inclined towards the center of the circle. So, all your forces must allow you to make this curved motion. So, when your road is designed as a curve, you must deliberately create a centripetal force, and then the object can make a centripetal motion. Many artistic designs also use this curved approach to design. So, you see, for example, the picture on the right is an artistic design, and it allows bicycles to display this beauty on it. So, you see, as long as it is a curve, it will definitely make a lean, and then create this centripetal force. So, it's not just the bicycle itself that leans, but the road surface also creates a lean, and then it generates centripetal force, allowing you to make this curved motion.