Chapter 559 257 Witness the Miracle! (Part 1)_3

So deep within Xu Yun's heart, he felt a bit guilty towards Wheat.

How to compensate Wheat later is another matter. In this process right now, what he can do is to get Wheat into the sight of these prominent figures as much as possible.

Of course.

Wheat did not know Xu Yun's thoughts. At this moment, he was holding a pen, scribbling down force analysis on the paper:

"Mr. Luo Feng said to disregard gravity, so it's only necessary to analyze the tension T at both ends of the segment AB."

"Segment AB is subjected to a tension T directed towards the lower left at point A and a tension T directed towards the upper right at point B, which are equivalent."

"However, the area of the segment is curved, so the directions of the two T's are not the same."

"Assume the angle between the tension at point A and the horizontal axis is θ, then the angle at point B with the horizontal axis is visibly different, marked as θ+Δθ."

"Because the points on the segment move up and down during oscillation, only the vertical component of the tension T needs to be considered."

"The upward tension at point B is T·sin(θ+Δθ), and the downward tension at point A is T·sinθ, so the net force on the entire segment AB in the vertical direction is the result of these two forces subtracted...."

Soon.

Wheat wrote down a formula on the paper:

F= T·sin(θ+Δθ) - T·sinθ.

Xu Yun nodded in satisfaction and said:

"Then what is the mass of the wave?"

"Mass of the wave?"

This time.

Wheat slightly furrowed his brows.

If we assume the mass per unit length of the segment is μ, then the mass of a segment of length Δl is obviously μ·Δl.

However, because Xu Yun selected a very small interval.

Assume the x-coordinate at point A is x, and the x-coordinate at point B is x+Δx.

This means the projected length of the rope AB along the x-axis is Δx.

So, when the selected rope length is very short and the wave is very small, Δx can approximately replace Δl.

Thus, the mass of the rope can be expressed as...

μ·Δx

Meanwhile.

Kirchhoff suddenly thought of something, his pupils slightly constricted, and he said in somewhat dry English:

"Wait a moment...both the net external force and mass have been determined, if we also derive the acceleration...."

Upon hearing Kirchhoff's words.

The classroom, which was not that noisy to begin with, suddenly quieted down even more.

Right.

Unknowingly, Xu Yun had already deduced the net external force and mass!

If the acceleration could also be deduced...

Wouldn't it be possible to express the equation of the wave in the classical system in the form of Niu Er?

Upon thinking this.

The prominent figures all took out paper and pen, tentatively calculating the final acceleration.

Speaking of acceleration, we must first talk about its concept:

This is a quantity used to measure the rate of change of velocity.

As for acceleration, the faster the velocity increases, the greater the value of acceleration.

For example, we often hear "I'm going to accelerate" and so on.

If a car's speed at the 1st second is 2m/s, and at the 2nd second is 4m/s.

Then its acceleration is the difference in speed (4-2=2) divided by the difference in time (2-1=1), resulting in 2m/s².

Recalling once again, how is the speed of a car determined?

Of course, it's the difference in distance divided by the difference in time that yields a value.

For instance, if a car is 20 meters from the starting point at the 1st second and 50 meters at the 2nd second.

Then its speed is the difference in distance (50-20=30) divided by the difference in time (2-1=1), resulting in 30m/s.

Have you noticed anything from these two examples?

Exactly!

Dividing the difference in distance by the difference in time gives the speed, and then dividing the difference in speed by the difference in time gives the acceleration, both processes involve dividing by the difference in time.

Then...

If you combine the two processes together?

Wouldn't it be possible to say:

By dividing the difference in distance by the difference in time once, and again by the difference in time, you get the acceleration?

Of course.

This is just a line of thought. Strictly speaking, this expression is not very precise, but it allows everyone to easily understand this concept.

If you consider distance as a function of time, then taking the derivative of this function once:

Is the above difference in distance divided by the difference in time, only tending toward an infinitesimal, thus obtaining a function of speed,

Taking the derivative of the function of speed once more gives you the representation of acceleration.

I don't know if the students of Xian Weiren understand, but all the prominent figures present quickly thought of this point.

Indeed.

The function f(x, t) listed earlier describes the position of a certain point on the segment at different times t!

So by taking the derivative of f(x, t) twice with respect to time, the acceleration a of this point is naturally obtained.

Since the function f is a function of two variables x and t, only partial derivatives with respect to time, ∂f/∂t, are possible, and taking the derivative again means adding a second order.

Hence soon.

Including Faraday, all the prominent figures successively wrote down a value:

Acceleration a= ∂²f/∂t².

Combining this value with the previously deduced net force and mass, a new expression emerged:

F= T·sin(θ+Δθ) - T·sinθ = μ·Δxa∂²f/∂t².

Then William Weber carefully looked at this expression, his brows furrowing slightly:

"Student Luo Feng, is this the final expression? I seem to feel like it can be further simplified?"

Xu Yun nodded:

"Of course it can."

F= T·sin(θ+Δθ) - T·sinθ = μ·Δxa∂²f/∂t².