每日彙整: 2018-04-08

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰運動學【二．六．四上】

假使有人說︰他知道

牛頓運動定律

牛頓運動定律（Newton’s laws of motion）描述物體與力之間的關係，被譽為是古典力學的基礎。這定律是英國物理泰斗艾薩克·牛頓所提出的三條運動定律的總稱，其現代版本通常這樣表述 ：[1]:88f[2]

  • 第一定律：存在某些參考系，在其中，不受外力的物體都保持靜止或等速直線運動。
  • 第二定律：施加於物體的淨外力等於此物體的質量與加速度的乘積。
  • 第三定律：當兩個物體互相作用時，彼此施加於對方的力，其大小相等、方向相反。

牛頓在發表於1687年7月5日的鉅著《自然哲學的數學原理》裏首先整理出這三條定律。[3]應用這些定律，牛頓可以分析各種各樣的動力運動。例如，在此書籍第三卷，牛頓應用這些定律與牛頓萬有引力定律來解釋克卜勒行星運動定律。

在鉅著《自然哲學的數學原理》1687年版本裡，以拉丁文撰寫的牛頓第一定律及牛頓第二定律。

 

尚不明白

Forces and moments/torques

 Forces are vectors which are applied to specific points (bound vectors) and moments/torques are vectors than describe rotational load applied to a body. Both can simply be described as vectors but either a point or reference frame must be associated with each, respectively.

 

Equal and Opposite

Don’t forget Newton’s third law of motion. If there is a force or torque, there is always an equal and opposit force or torque acting on the opposing point or reference frame.

 

Equations of motion

 

Once all of the important forces acting on a system, the accelerations of all particles and bodies, and the inertial properties of the system are found, the equations of motion can be formed. For planar dyanmics the equations take on this form:

\sum \vec{F} = m \vec{a}

\sum \vec{T} = I \vec{\alpha}

The force equation (Newton’s second law) and the torque equation (Euler’s equation) make up a set of second ordinary differential equations in time. We typically want these equations in first order form:

\dot{x} = f(x, u, t)

And to do that kinematical differential equations are introduced, which simply define the relationships between the positional and angular states and their derivatives. For example, we could introduce ω as the time derivative of an angle θ:
\omega = \dot{\theta}
The states xx are typically positions, angles, and their rates.
 

In general, the equations of motion are non-linear ordinary differential equations and anayltical solutions do not exist. To find the resulting state trajectories we turn to numerical integration methods.

x = \int_{t_0}^{t_f} f(x, u, t) dt

 

文本之論述概括也！恐不過是思考還沒經過『力學分析』鍛鍊，未能深入了解耳◎

舉例而言如何用『科學原理』闡釋『竹蟬』童玩現象呢？

國立台中教育大學 NTCU

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竹蟬

「知了」的叫聲很響亮，紙杯可以做成知了嗎？

 

就算現已是春天，不管知了鳴聲叫響。僅就『轉動』一事而言，仍需鋪陳演繹︰

Circular motion

In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.

Examples of circular motion include: an artificial satellite orbiting the Earth at a constant height, a fan’s blades rotating around a hub, a stone which is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.

Since the object’s velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, according to Newton’s laws of motion.

Figure 1: Velocity v and acceleration a in uniform circular motion at angular rate ω; the speed is constant, but the velocity is always tangent to the orbit; the acceleration has constant magnitude, but always points toward the center of rotation.

Figure 2: The velocity vectors at time t and time t + dt are moved from the orbit on the left to new positions where their tails coincide, on the right. Because the velocity is fixed in magnitude at v = r ω, the velocity vectors also sweep out a circular path at angular rate ω. As dt → 0, the acceleration vector a becomes perpendicular to v, which means it points toward the center of the orbit in the circle on the left. Angle ω dt is the very small angle between the two velocities and tends to zero as dt→ 0.

 

以及邏輯推導︰

Uniform circular motion

In physics, uniform circular motion describes the motion of a body traversing a circular path at constant speed. Since the body describes circular motion, its distance from the axis of rotation remains constant at all times. Though the body’s speed is constant, its velocity is not constant: velocity, a vector quantity, depends on both the body’s speed and its direction of travel. This changing velocity indicates the presence of an acceleration; this centripetal acceleration is of constant magnitude and directed at all times towards the axis of rotation. This acceleration is, in turn, produced by a centripetal force which is also constant in magnitude and directed towards the axis of rotation.

In the case of rotation around a fixed axis of a rigid body that is not negligibly small compared to the radius of the path, each particle of the body describes a uniform circular motion with the same angular velocity, but with velocity and acceleration varying with the position with respect to the axis.

Formulas

Figure 1: Vector relationships for uniform circular motion; vector ω representing the rotation is normal to the plane of the orbit.

For motion in a circle of radius r, the circumference of the circle is C = 2π r. If the period for one rotation is T, the angular rate of rotation, also known as angular velocity, ω is:

\displaystyle \omega ={\frac {2\pi }{T}}=2\pi f={\frac {d\theta }{dt}} and the units are radians/second

The speed of the object travelling the circle is:

\displaystyle v={\frac {2\pi r}{T}}=\omega r

The angle θ swept out in a time t is:
\displaystyle \theta =2\pi {\frac {t}{T}}=\omega t\,
The angular acceleration, α, of the particle is:
\displaystyle \alpha ={\frac {d\omega }{dt}}
In the case of uniform circular motion, α will be zero.

The acceleration due to change in the direction is:

\displaystyle a={\frac {v^{2}}{r}}=\omega ^{2}r

The centripetal and centrifugal force can also be found out using acceleration:
\displaystyle F_{c}={\dot {p}}\ {\overset {{\dot {m}}=0}{=}}\ ma={\frac {mv^{2}}{r}}
The vector relationships are shown in Figure 1. The axis of rotation is shown as a vector ω perpendicular to the plane of the orbit and with a magnitude ω = dθ / dt. The direction of ω is chosen using the right-hand rule. With this convention for depicting rotation, the velocity is given by a vector cross product as
\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} \ ,
which is a vector perpendicular to both ω and r(t), tangential to the orbit, and of magnitude ω r. Likewise, the acceleration is given by
\displaystyle \mathbf {a} ={\boldsymbol {\omega }}\times \mathbf {v} ={\boldsymbol {\omega }}\times \left({\boldsymbol {\omega }}\times \mathbf {r} \right)\ ,
which is a vector perpendicular to both ω and v(t) of magnitude ω |v| = ω2 r and directed exactly opposite to r(t).[1]

In the simplest case the speed, mass and radius are constant.

Consider a body of one kilogram, moving in a circle of radius one metre, with an angular velocity of one radian per second.

 

的呦！☆

如是能否解釋 『向心力概念』之意指乎？★

向心力

在古典力學中，向心力是當物體沿著圓周或者曲線軌道運動時，指向圓心（曲率中心）的合外力作用力。「向心力」一詞是從這種合外力作用所產生的效果而命名的。這種效果可以由彈力、重力、摩擦力等任何一力而產生，也可以由幾個力的合力或其分力提供。

因為圓周運動屬於曲線運動，在做圓周運動中的物體也同時會受到與其速度方向不同的合外力作用。對於在做圓周運動的物體，向心力是一種拉力，其方向隨著物體在圓周軌道上的運動而不停改變。此拉力沿著圓周半徑指向圓周的中心，所以得名「向心力 」。向心力指向圓周中心，且被向心力所控制的物體是沿著切線的方向運動，所以向心力必與受控物體的運動方向垂直，僅產生速度法線方向上的加速度。因此向心力只改變所控物體的運動方向，而不改變運動的速率，即使在非勻速圓周運動中也是如此。非勻速圓周運動中，改變運動速率的切向加速度並非由向心力產生。

向心力的大小與物體的質量（m）、物體運動圓周半徑的長度（r）和角速度（ω）有著密切關係。