After taking the advice, I became a top student.

Chen Mo was unaware that a force had been watching him from behind; at this moment, he was completely absorbed in his studies.

Listening to the lecture, the slight arrogance that hadn't even had a chance to grow in Chen Mo's heart completely disappeared. Compared to university physics, high school physics was like a fallen leaf in a vast forest.

Yet, some people can effortlessly and skillfully explain complex physical concepts in a clear, easy-to-understand, vivid, and interesting way, despite the vast subject of university physics.

The outside world is indeed full of powerful people; there are always people more capable than you!

Soon, the physics class ended, and Chen Mo walked up to the podium with his textbook the moment he heard the bell.

"Teacher Wang, there's a question I haven't understood. Could you help me with it?"

Chen Mo said sincerely that he wasn't just there to fulfill a suggestion; after studying university physics, he really felt there were still many things he didn't understand.

"Oh?"

"What question?"

Wang Xiaohua nodded in satisfaction. As long as Chen Mo was still willing to study hard, he was a good child.

That's right.

Chen Mo took his draft paper, on which he had copied the questions.

A uniform solid cylinder of mass m and radius R is released from rest from an inclined plane with an angle of θ. The inclined plane is rough enough that the cylinder initially rolls without slipping. The coefficient of static friction between the inclined plane and the cylinder is μs, and the coefficient of kinetic friction is μk, where μk < μs.

1. Find the acceleration of the center of mass and the angular acceleration of the cylinder during the pure rolling phase.

2. As the cylinder's velocity increases, it may start to slide at some point. Find the condition under which it starts to slide.

3. If the cylinder does indeed enter the sliding phase after rolling a certain distance, find the velocity and angular velocity of the cylinder's center of mass after time t from the start of the sliding phase.

Wang Xiaohua frowned.

This is not high school physics at all.

She even suspected that Chen Mo had deliberately chosen a trap-filled question to trick her.

But she was, after all, a top student who graduated from Central China Normal University and had taught high school physics for decades. Her foundation in university physics was still there, and the moment she saw the question, she keenly sensed the problem.

"This problem looks like a standard pure rolling problem..."

Wang Xiaohua decided to feign innocence and outsmart this cunning student, asking, "So, what's the part you don't understand?"

Chen Mo immediately added: "But what I find strange is that many materials only discuss the critical conditions for starting to slide, and rarely specifically calculate the speed after sliding."

Furthermore, if we consider that the moment of inertia of the cylinder is 1/2mR^2, will the direction of the frictional force change after sliding?

Also, if the angle of inclination is very small, is it possible for the cylinder to roll continuously? If the angle of inclination is large, will it slide from the beginning? How is this transition point strictly determined?

"Oh?"

Wang Xiaohua was indeed somewhat surprised this time.

She initially thought Chen Mo was pretending to know what he was talking about, deliberately finding a tricky college physics question to cause trouble, but as soon as Chen Mo opened his mouth, she knew that this little guy seemed to be seriously thinking about the question.

The trap in this question lies in the premise of the transition from rolling to sliding. Under normal circumstances, a cylinder either rolls continuously or slides from the beginning; there is no scenario where it starts sliding midway through rolling.

Chen Mo's question precisely points out this subtle point, requiring in-depth analysis of the changes in static friction during pure rolling.

This requires taking into account the changes in the initial angular velocity of the cylinder and the coefficient of friction of the inclined plane with position.

"Seriously, buddy, you ask questions in every single class? Where do you get so many questions?"

"Before, I only asked math questions, but now I ask them in every class. I'm amazed at how many questions they have."

"Now that you mention it, I remember. He asked me questions about math before, and then got a perfect score on the midterm math test. Now he's starting to ask questions about other subjects. I hope he doesn't get perfect scores in all the other subjects next time, right?"

"You're dreaming!"

The students whispered among themselves, thinking Chen Mo was a conspicuous figure.

"That's a very good question!"

On the podium, Wang Xiaohua nodded and stopped trying to outsmart the student. After thinking seriously for a while, she turned around and began to carefully derive the solution on the blackboard.

"First, we apply Newton's second law to the center of mass: mgsinθ−f=ma_c. Then we apply the rotational law to the center of mass: fR=I_cα, where Ic=1/2mR^2. The pure rolling condition is ac=αR. Solving these equations simultaneously, we get: ac=2/3gsinθ, f=1/3mgsinθ."

"Therefore, we can obtain the condition for the sliding to begin by substituting static friction: tanθ≤3μs."

"That is to say, when tanθ>3μs, the static friction is not enough to maintain pure rolling, and the cylinder will slide from the beginning."

When tanθ < 3μs, the cylinder can maintain pure rolling for a period of time.

However, as the speed increases, if the frictional force does not reach the maximum static friction, the cylinder may continue to roll to the bottom.

In fact, on a uniform inclined plane, f is constant during the pure rolling phase. Therefore, if tanθ < 3μs, the cylinder will roll continuously to the bottom without sliding midway.

However, if other factors are considered, such as different initial conditions of rotation, the results may change.

This problem assumes that the cylinder starts from rest and rolls purely. Then, when tanθ < 3μs, it will not slide.

Therefore, the condition for the sliding to begin is tanθ > 3μs, and at this point, the cylinder is simultaneously sliding and rolling from the very beginning, rather than transitioning from pure rolling.

After Wang Xiaohua finished the derivation, she casually tossed the chalk into the chalk box. She felt that she had solved the problem perfectly and was quite proud of herself.

Chen Mo, standing to the side, suddenly realized that the problem was not difficult. As long as one knew the formula and theorem, any junior high school student could derive the formula.

However, analyzing the entire process requires a solid foundation in physics and physical intuition; otherwise, one may easily fall into the trap of the question and arrive at the wrong answer.

He felt deeply that he still knew too little and needed to learn more.

"I'm so glad you like physics, but you're still a high school student, so you should focus on your studies. Once you get into university, you can study whatever you like to your heart's content."

Wang Xiaohua looked at Chen Mo's enlightened expression, and it seemed that he really understood. This could indeed help the boy's talent, but she still hoped that Chen Mo could spend more energy on high school physics, since the current trend is still to prioritize college entrance examination scores.

"Zhou Hao, what problem were Teacher Wang and Chen Mo discussing? I don't quite understand."

In the classroom, most of the students were captivated by Wang Xiaohua and her companion's actions. They stared at the scribbles on the blackboard; they recognized each letter individually, but when put together, they began to feel dizzy.

Zhou Hao stared at the blackboard, remaining silent.

I don't understand this either!

"I suggest that Chen Mo ask questions in the office from now on, instead of torturing us slackers in the classroom."

Huang Qinghua stared blankly at the blackboard and complained to Li Ze beside him.

Guo Chao exclaimed in astonishment, "Is this the language of the academic world? I can't understand a single word!"

"It would be strange if you understood. Don't you know there's a reproductive barrier between top students and underachievers?"

Huang Qinghua delivered a scathing rebuke.

Ring ring ring...

The school bell had already rung, and math teacher Hu Xin had arrived at the classroom door.

When he saw the series of formulas on the blackboard, he was slightly stunned. Are high school physics teachers nowadays teaching such profound knowledge?

He probably remembers learning it when they were in college.

They really are getting better with each generation!