Free Web NovelChapter 52: Chapter 49: Almost Forgot to Fleece the Sheep
The teachers’ office for the elite class at Jiangcheng Sixth Middle School was different from the subject-and-grade-level group model found in other schools.
Inside the spacious office sat only the top teachers from each subject, all responsible for teaching the elite class.
This layout was designed to make it convenient for the teachers to conduct comprehensive performance analyses and joint assessments of the several dozen top students in the class at any time.
Such an absolute concentration of teaching staff and allocation of resources was unimaginable in an ordinary public high school.
This was also why Liu Limin dared to pat his chest and declare that the platform Sixth Middle School offered could provide Li Dong with a higher ceiling.
When Elder Chen led Li Dong through the office door, no other teachers were inside.
He walked to his desk, pulled open a drawer, and began rummaging through it.
"What kind of problem should I find for this kid?"
Elder Chen muttered to himself.
"Using Calculus to derive an electromagnetic oscillating circuit? No..."
Elder Chen had just pulled out a sheet of paper when he remembered that Li Dong was a monster who could mentally compute double integrals and partial derivatives. He shook his head decisively and stuffed the paper back in.
When it came to mathematical calculation tools, this kid was simply a freak.
’I need to find a problem that can truly test his physical intuition.’
Finally, he pulled an A4 sheet of paper from a folder.
He looked at the problem on the paper and nodded.
The problem wasn’t actually that difficult. For a student who regularly participated in physics competitions, as long as they had a bit of talent, they could eventually grind out the answer.
Of course, that also depended on the school.
If this were at Jiangcheng Seventh Middle School... it would definitely be enough to stump Jiang Yizhou and make him cry on the spot.
This problem required inspiration, a specific angle of approach.
If you had sharp physical intuition and could see through to the correct physical picture hidden in the problem at a glance, you could solve it in seconds.
But if you couldn’t find that entry point and tried to brute-force it with conventional dynamics...
...then congratulations. Even if you filled ten pages with Lagrangian equations, you would end up with a nonlinear differential equation with no elementary analytic solution, leading you down a dead end you could never escape.
In the end, your physics teacher would just pat your shoulder and earnestly advise you, "Kid, maybe physics isn’t the path for you."
Elder Chen handed the A4 paper to Li Dong.
"Here, try this."
Li Dong took it curiously and glanced down.
[A uniform, thin, rigid rod of mass m and length 2l initially stands vertically on a smooth horizontal tabletop. Due to a small perturbation, the rod begins to fall. Find: during the process of falling while the rod remains in contact with the tabletop, the trajectory equation of its instantaneous center of rotation (the instantaneous center), and sketch the trajectory curve.]
After reading the problem, Li Dong raised an eyebrow slightly.
"Huh? This one’s a little interesting."
The moment he saw the problem, he recognized its trap.
If an ordinary high school student, or even a college freshman, saw this kind of rigid body motion, their first instinct would absolutely be to establish a coordinate system!
Then, they would list the translational equations for the center of mass, write out the rotational equations around the center of mass, and finally try to force a solution for the coupled differential equations.
But once you actually did that, you would discover in despair that the relationship between the rod’s angle θ and time t as it falls is a complex nonlinear function, making it impossible to find an elementary analytic solution!
Your final scratch paper would be an absolute mess.
Therefore, this problem wasn’t a test of computational ability at all!
In Li Dong’s mind, the imaginary rigid rod had already begun to fall on the smooth tabletop.
’The horizontal surface is a smooth tabletop, which means there are no external horizontal forces!’
’Since there are no horizontal external forces, the momentum of the rod’s center of mass in the horizontal direction must be conserved! The initial momentum is zero, so the momentum throughout the process is also zero.’
’Therefore, the center of mass cannot have any horizontal displacement. It can only fall straight down along a vertical line!’
Li Dong’s eyes lit up.
’Isn’t this just a classic sliding ladder model?’
The entry point—found!
’This Teacher Chen really wasn’t kidding. This problem is actually pretty fun.’
Since it was so much fun, Li Dong decided to derive the trajectory of the instantaneous center from three different perspectives.
He picked up his pen and wrote "Solution 1" and "Solution 2" on the paper. Just as he was about to write "Solution 3"...
CLICK.
The office door was pushed open.
A refined-looking teacher walked in, holding a thick stack of test papers, and casually placed them on a desk.
Li Dong subconsciously glanced up.
It was the biology teacher from the previous class!
The moment he saw the biology teacher, Li Dong’s mind went blank for a second, as if he had forgotten something important.
’Holy shit!’
’Mendel!’
’The "three-line system" for cytoplasmic male sterility that the biology teacher talked about... I still haven’t completely figured out the underlying logic!’
’Even worse, Mendel, the guy who grew peas, is in the "Cyan Dragon Study Group" right now, eagerly waiting for my explanation!’
Li Dong suddenly realized he had no time to waste!
’Those titans in the group could go offline at any moment.’
’If Mendel goes offline, won’t this whole goddamn haul of wool just fly away?!’
’Compared to a god-tier haul of wool, what’s more important: the instantaneous center trajectory of this stupid rod?’
’Multiple solutions for one problem? A detailed derivation process? Elegantly showing off in front of the teacher?’
’To hell with all of it!’
Li Dong immediately started writing on the A4 paper.
Meanwhile, Elder Chen was leisurely unscrewing the lid of his thermos.
He then glanced at the time, preparing to time Li Dong.
’If this little guy can see through it in under a minute, then he’s got some exceptional talent.’
’If he can do it in thirty seconds... then this kid’s physical intuition is good enough to try for the Ossai.’
Elder Chen had just raised his thermos, ready to take a pleasant sip of hot water.
However, the hot water hadn’t even touched his lips.
"Teacher, I’m done. I have a question about a concept for the biology teacher. Please wait for me a moment!"
Elder Chen froze. His hand jerked, and a few drops of tea splashed onto the back of his hand.
Then he saw a figure dash past him.
"What the hell was that?"
He quickly put down his cup and looked up. Li Dong had already run over to the biology teacher.
"He’s... done already?"
Elder Chen hurriedly looked down at the A4 paper Li Dong had just placed on the desk.
The handwriting on the paper was messy.
[Solution 1:
As there are no external forces in the horizontal direction, the momentum of the center of mass C is conserved horizontally. Thus, the center of mass C only undergoes vertical downward linear motion.
Let the rod’s point of contact with the surface be A, and the center of mass be C.
From the method for finding the instantaneous center of a rigid body in planar motion, the instantaneous center I must be at the intersection of the line passing through point A perpendicular to its direction of motion and the line passing through point C perpendicular to its direction of motion.
That is, the instantaneous center I is the vertex of the right angle in the right triangle ACI.
In this right triangle ACI, the length of the hypotenuse AC is always constant, equal to the semi-rod length l.
Establish a rectangular coordinate system with the origin at the rod’s initial point of contact (the intersection of the tabletop and the vertical line of the center of mass’s motion). Regardless of the angle θ through which the rod has fallen, based on the geometric relationship, the coordinates (x, y) of the instantaneous center I will always satisfy the Pythagorean theorem for a right triangle:
x² + y² = l²]
[Answer: The trajectory of the instantaneous center is a quarter-circular arc with a radius of l.]
Not a single wasted word.
He had directly seen through the essence of the falling rigid body from the highest dimension of kinematic geometry.
Looking at these few lines, Elder Chen wasn’t surprised that Li Dong could solve it.
But the time it took...
Including the time to read the problem, think, and write.
All told, less than a minute?
Then he looked at the two words "Solution 2" next to "Solution 1".
What did this mean?
It meant that the instant this kid saw the problem, not only did he instantly figure out the simplest geometric approach, but he had also simultaneously conceived of a second one in his head!
’Two solution methods?’
Elder Chen took off his glasses, wiped the lenses with the corner of his shirt, and put them back on.
He looked over at Li Dong, who was eagerly questioning the biology teacher nearby, and couldn’t help but smile.
"What a little monster."
