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Don't you think it's too easy to see this topic? On the face of it, the problem is very simple, but many people do not understand it or even misunderstand it completely.

As a physics term, rolling is not simple. To understand it, we need to start with the two basic forms of rigid body motion, which are translation and rotation.

Tip: because the particle can only be translated, it is not considered, so the "object" in this paper refers to the "rigid body", that is, the object whose deformation can be ignored.

Translation means that the object does not move; rotation means that the object does not move.

What? That sounds a little dizzy.

It's a bit of a mess, though it's right to say so! But in fact, they all have strict definitions!

Translation: the connection between any two points inside an object. If the line is parallel all the time during the motion of the object, the motion is translational.

Like this.

Rotation: any point on an object moves in a circle around a given point.

It's simple, that's it.

The two of them are combined in different proportions to form a variety of different sports!

If this combined motion occurs on a face (plane or surface), it is habitually called scrolling.

It looks easy, right?

Let's start with the most typical rolling, pure rolling, which refers to a roll without slipping or skidding, or a combination of rotation and translation at 1:1.

Maybe you're a little confused about the word "no skid"?

In fact, you may think too much, you just need to take it literally. If you don't slip, you just don't move! The movement is relative, and the motion here is relative to the ground, so not skidding means not moving relative to the ground-boy, isn't it moving? How can it not move to the ground?

The term "motionless" here means that "the part that is in contact with the ground is motionless relative to the ground". For example, when you walk, the foot you step on the ground actually does not move, but the other foot off the ground.

Take the wheel as an example, if it rolls to the ground, then the point where it touches the ground (the lowest point of the wheel) is relatively motionless! Let's analyze it carefully.

Before making a formal analysis, let's review the relativity of motion in classical mechanics.

If there are three particles A, B and C, then the relativity of their velocities can be expressed as "the former versus the latter", for example, the speed of A relative to B.

The second letter of the subscript in the above speed symbol represents the reference, for example, "the speed of A when B is selected as the reference".

The relativity of velocity shows that any relative velocity can always be regarded as the sum of many other relative velocities.

Now let's look at the speed of the lowest point of the wheel.

As shown in the following figure, the wheel is rolling, let its radius be, and the angular velocity around the center is.

Let the speed of the lowest point of the wheel, the speed of the lowest point relative to the center point, and the speed of the center point relative to the ground are expressed as, and, respectively.

Obviously, the direction is horizontal to the left, and the direction is horizontal to the right. As for the direction is uncertain, it is temporarily indicated by a signed number. Positive to the right, the relative relationship between the three velocities is

According to the relationship between linear velocity and angular velocity, the speed of the lowest point of the wheel relative to the center of the wheel is horizontal to the left. So the above formula can be written as the above meaning of non-skidding, the lowest point of the wheel speed is zero, so there must be

This is the relationship between the wheel center speed and the wheel speed when the wheel is rolling.

In fact, not only the central speed of the wheel satisfies this relationship, for a pure rolling wheel, the speed of any point to the ground is equal to the angular velocity of the wheel multiplied by the distance between that point and the contact point on the ground. For example, the speed at the highest point of the wheel is zero and the speed at the foremost point is zero.

This is easy to understand. Since the lowest point of the wheel does not move, it is equivalent to the ground. The relative speed of any point on the wheel is its speed relative to the lowest point, and its motion relative to the lowest point is the rotation with the corresponding distance as the radius.

That's right! The speed of any point to the ground is the distance from that point to the lowest point.

But you may have a little doubt, is this the angular velocity of the wheel around the center?

Right! It can be proved that no matter which point on the wheel, the angular speed of the wheel is the same. This is called the absoluteness of angular velocity of a rigid body. There is no proof here. If you are interested, you can read the book by yourself.

At this point, do you realize that the speed of each point on the rolling wheel is different?

That's right! Because the bicycle wheel in motion is rolling, not turning!

Experienced people will notice that this is why the spokes look blurred as they move up the wheels of a bicycle.

With regard to the pure rolling of a wheel, an equivalent term was mentioned earlier, which is "the combination of rotation and translation at 1:1". What does that mean?

It means that the speed of any point on the edge of the wheel is the superposition of the translational speed of the wheel as a whole and its rotation speed relative to the center of the wheel at the ratio of 1:1.

It's obvious!

Because the wheel as a whole translates with all the points, the speed is horizontal, and the speed of any point on the edge of the wheel relative to the center is tangential. According to the relativity of velocity, their sum is the velocity of that point. As has been proved before, the two speeds are equal when the wheels do pure rolling. This shows that the speed at any point on the edge of the wheel is always equal to the synthesis of two equal speed vectors! This is what the 1:1 ratio means.

In fact, pure rolling is not limited to wheels. If you look at the motion of these objects below, it is also obvious that they are pure rolling-because the velocity of their points of contact with the ground is zero, and the speed of each point on the object is satisfied.

After talking about pure scrolling, let's look at non-pure scrolling.

Very simply, non-pure rolling is a slip at the point of contact, that is, the speed of the lowest point of the wheel, so for non-pure rolling, the speed on the edge of the wheel is not combined according to translation and rotation at 1:1. And for any point on any rigid body, non-pure rolling means.

If further analysis, the non-pure scrolling is divided into two cases according to greater or less than.

The right situation, which often occurs when the car is braking, is also called slip, which is translated as slipping. In order to describe the degree of slip, the definition of slip rate is as follows, which often occurs when the car starts at acceleration, also known as skidding. Similarly, the definition of slip rate is self-evident.

For example, when your suitcase wheels stop turning, you have to drag hard, and the slip rate is 100%, and your suitcase brakes are too good; when your wheels are stuck in the quagmire, the wheels spin fast, but the car doesn't move at all. The slip rate is 100%.

That's all I have to say about rolling.

If the motion of a rigid body is not limited to one surface, it is not called rolling, but the general motion of a rigid body. The general motion of a rigid body is complex, which is not shown here for the time being.

This article is from the official account of Wechat: University Physics (ID:wuliboke), by Xue Debao.

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