Mechanical problems for experts

Are you analyzing the specific problem in detail? Is this problem difficult? It is very easy.

Do you think it is necessary to consider the energy used to rotate this piece, unless you use energy in the whole process? To solve,

Due to the conditions here, I will only talk about the idea

The first step: first determine whether there is sliding. If there is no sliding, the mechanical energy is conserved in the whole process. That is too simple. Just be excited. If your judgment slips, you will be even more excited. The challenge is here and you can show off, but it is still at the rookie level. Smile to yourself.

Step 2: In the case of sliding, the whole process is divided into two parts, from the coexistence of rolling and sliding to the coexistence of rolling only. The more complicated stage is the first stage, that is, the stage of coexisting rolling and sliding. The critical state at the connection point between the coexistence of rolling and sliding and pure rolling must be that the linear speed of the cylinder's edge rotation is equal to its translation speed, that is, the center of mass speed. So we get the critical condition, angular velocity * radius = center of mass translation velocity.

Let’s express the angular velocity first. Since it is sliding friction, it is obvious that the friction force in the first stage is undoubtedly a constant force. Since gravity acts on the center of mass, and the center of mass is on the axis of the cylinder, gravity exerts an influence on the cylinder. There is no contribution from body rotation, so angular acceleration comes entirely from sliding friction. Therefore, the moment can be found. The moment is divided by the moment of inertia, and the angular acceleration is obtained. The angular acceleration multiplied by time is the angular velocity of rotation. The time here is unknown, this is an indirect intermediate variable we require. We have an expression for angular velocity, and the only unknown quantity is time.

Let’s express the translational speed of the cylinder. Friction and gravity will obviously contribute to the acceleration. I don’t need to say more about how to find this acceleration. This is the basic skill of basic skills, something at the kindergarten level. , then isn’t this acceleration*time the translation speed? Obviously this time is equal to the time above, and there is only one unknown quantity: time.

Using what we said, angular velocity * radius = center of mass translation velocity. We can find the exercise time. Why do you ask for this time? It seems to have nothing to do with the topic. Let's continue to break it down.

At this point we have fully understood the motion state at the end of the first stage. We enter the third step.

Step 3: If you have been thinking about it, you may think that we have already dealt with the first stage of motion and should deal with the second stage, which is the pure rolling stage, but doing so will actually cause problems. It’s complicated. We don’t consider the movement in the second stage, but combine the two stages into one and look at the whole process. What should we do? Here we use the conservation of energy, (not using the kinetic energy theorem, nor the conservation of mechanical energy, because motion is divided into two stages. In the first stage, mechanical energy is not conserved, so we can only use the kinetic energy theorem, while in the second stage, mechanical energy is conserved, and the whole process is variable force. So consider the conservation of energy)

Due to the conservation of energy, how much energy is in the initial state of motion, then the energy he can use will not decrease or increase, that is, the initial kinetic energy is the total energy. During the motion , the energy loss comes from sliding friction. When reaching the highest speed, the speed is zero, so there is only potential energy.

So: total energy = mechanical energy loss of sliding friction + gravitational potential energy of the final state. We call this equation the terminator of the problem. That is the terminator equation

The total energy is the initial kinetic energy, so it is easy to handle. Then the gravitational potential energy is mass times height. Obviously the height is unknown. Is this what we require? As for the energy consumption of sliding friction, you can figure it out with your brain. We know the translational acceleration of the cylinder, the initial speed, and the movement time. Then we know the distance of movement on the slope in the first stage of the rolling stage. Friction is a constant force, so the work it does can be calculated. The only mechanical energy loss in the whole process is the mechanical energy loss, so the mechanical energy loss can be calculated. Now the mechanical energy loss has become a known quantity. Therefore, the total energy becomes a known quantity, the mechanical energy loss becomes a known quantity, and only the height of the gravitational potential energy term is unknown, so our Terminator equation only has the unknown quantity of height.

Here I will list and analyze all possible situations. The first one is no sliding. We have already mentioned the first step, it is simple.

The second one is that there is sliding, so we divide the motion into two stages: rolling and sliding coexist, and the pure rolling stage. Is it possible that the pure rolling stage does not exist? That is impossible, because when there is rolling to pure rolling, The critical condition for rolling is that the linear velocity of rotation is equal to the center of mass velocity. If the second stage does not exist, it means that the center of mass velocity is zero at the end of the first stage. But can the angular velocity be zero at this time? It is obviously impossible. In the first stage, the direction of the friction force has never changed, and the direction of the angular acceleration is constant. It does not change, that is, the angular velocity has been increasing during this process, starting from zero. By the end of the first stage, the angular velocity should be the maximum, and it cannot be zero. Therefore, as long as there is a rolling phase, there must be a pure rolling phase.

So far we have solved this problem. It is very simple. Now we can be more excited. The problem is solved.

Let’s talk about the mistakes above. Everyone has thought of using energy conservation, but the critical state is not clearly distinguished. The most obvious mistake is that the first floor believes that there is only the rolling stage and no pure rolling stage. Therefore, the distance of work done by friction is wrong, so the calculated height On the small side. The second floor, no doubt, does not lack conditions. The third floor is the same as the first floor. I don't see that the movement process is divided into two stages. The fourth floor has the same idea as the first and third floors, and the mistakes are the same, but I don't understand the formula very well.

Take a closer look at the movement process, as long as you can distinguish two possible situations of movement, and the second situation must have two stages. The problem becomes very simple. In addition, I would like to add that this is a practice question for physics competition training, and it is a relatively simple one. In the moment of inertia, the conservation part of angular momentum is associated.