简单调和动作

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A Foucault is a pendulum suspended from a long wire, that is sustained in motion over long periods. Due to the axial rotation of the Earth, the plane of motion of the pendulum shifts at a rate and direction dependent on its latitude: clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere. At the poles the plane rotates once per day, while at the equator it does not rotate at all.
::A Foucault 是一个从长线上悬浮的钟摆,长线长长,长线长长。 由于地球轴心旋转,钟摆移动的平面,其速度和方向取决于其纬度:北半球顺时针,南半球逆时针。在极上,飞机每天旋转一次,而在赤道则完全不旋转。

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Simple Harmonic Motion
::简单调和动作

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Many objects vibrate or oscillate – an object on the end of a spring, a tuning fork, the balance wheel of a watch, a pendulum, the strings of a guitar or a piano. When we speak of a vibration or oscillation, we mean the motion of an object that repeats itself, back and forth, over the same path. This motion is also known as simple harmonic motion , often denoted as SHM.
::许多物体振动或振动 — — 弹簧末端的物体、调音叉、钟表的平衡轮、钟表、吉他或钢琴的弦。 当我们谈论振动或振动时,我们指的是一个在同一个路径上前后重复的物体的动作。 这个动作也被称为简单的调音动作,通常被称为SHM。

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A useful design for examining SHM is an object attached to the end of a spring and laid on a surface. The surface supports the object so its (the force of gravity) doesn’t get involved in the forces. The spring is considered to be weightless.
::用于检查SHM的有用设计是连接在弹簧末端并埋在表面的物体。 表面支撑该物体,使其(重力)不卷入力中。 弹簧被认为是无重力的。

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The position shown in the illustration is the equilibrium position . This position is the middle, where the spring is not exerting any force either to the left or to the right. If the object is pulled to the right, the spring will be stretched and exert a restoring force to return to the weight to the equilibrium position. Similarly, if the object is pushed to the left, the spring will be compressed and will exert a restoring force to return the object to its original position. The magnitude of the restoring force, $\begin{array}{r}F\end{array}$ , in either case must be directly proportional to the , $\begin{array}{r}x\end{array}$ , the spring has been stretched or compressed. (A spring must be chosen that obeys this requirement.)
::插图中的位置是平衡位置。 此位置是中间, 弹簧没有向左或向右施加任何力量。 如果弹簧被拉到右边, 弹簧将被拉伸, 并施加恢复力返回到平衡位置的重量。 同样, 如果弹簧被推到左边, 弹簧将被压缩, 并施加恢复力, 使物体返回到原来的位置 。 恢复力的大小 F , 无论哪种情况, F 都必须与弹簧被拉长或压缩过直接成比例 。 (必须选择符合这一要求的弹簧 。)

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$\begin{array}{r}F=-kx\end{array}$
::Fkx

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In the equation above, the constant of proportionality is called the spring constant . The spring constant is represented by k and its units are N/m. This equation is accurate as long as the spring is not compressed to the point that the coils touch nor stretched beyond elasticity.
::在以上方程中,相称性常数被称为春季常数。春季常数由 k 表示,其单位为 N/m。只要弹簧不压缩到圆圈触碰或伸展超过弹性的点,此等值是准确的。

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Suppose the spring is compressed a distance $\begin{array}{r}x=A\end{array}$ , and then released. The spring exerts a force on the mass pushing it toward the equilibrium position. When the mass is at the maximum displacement position, is zero because the mass is changing direction. At the position of maximum displacement, the restoring force is at its greatest - the of the mass will be greatest. As the mass moves toward the equilibrium position, the displacement decreases, so the restoring force decreases and the acceleration decreases. When the mass reaches the equilibrium position, there is no restoring force. The acceleration, therefore, is zero, but the mass is moving at its highest velocity. Because of its , the mass will continue past the equilibrium position, and stretch the string. As the spring is stretched further, the displacement increases, the restoring force increases, the acceleration toward the equilibrium position increases, and the velocity decreases. Eventually, when the mass reaches its maximum displacement on this side of the equilibrium position, the velocity has returned to zero and the restoring force and acceleration have returned to the maximum. In a frictionless system, the mass would oscillate forever, but in a real system, gradually reduces the motion until the mass returns to the equilibrium position and motion stops.
::假设弹簧压缩了一个距离 x=A, 然后释放。 弹簧在质量上施压, 将它推向平衡位置。 当质量处于最大迁移位置时, 质量是零, 因为质量正在改变方向。 在最大迁移位置, 恢复力是最大—— 质量将是最大的。 当质量向平衡位置移动时, 迁移力会下降, 因此恢复力会下降, 加速力会下降。 当质量到达平衡位置时, 没有恢复力。 因此, 加速力是零, 但质量会以最高的速度移动。 由于其 , 质量会持续过平衡位置, 并拉伸绳子。 随着弹簧的伸展, 迁移力会增加, 恢复力会增加, 向平衡位置加速, 速度会下降。 最终, 当质量到达平衡位置的这一侧部位时, 速度会恢复到零, 恢复力和加速速度会恢复到最大。 在没有摩擦的系统中, 质量会永远流动, 但是在一个实际系统中, 逐渐减少运动, 直到大规模恢复到平衡位置和运动。

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Imagine an object moving in uniform . Remember the yo-yo we spin over our heads? In your mind, turn the circle so that you are looking at it on edge; imagine you are eight feet tall, and the yo-yo's circle is exactly at eye level. The object will move back forth in the same way that a mass moves in SHM. It moves consistently from the far left to the far right until you stop spinning the yo-yo. Another example is to imagine a glowing light bulb riding a merry-go-round at night. You are sitting in a chair at some distance from the merry-go-round so that the only part of the system that is visible to you is the light bulb. The movement of the light will appear to you to be back and forth in simple harmonic motion. Circular motion and simple harmonic motion have a lot in common.
::想象一个穿着制服移动的物体 。 记得我们旋转在我们的头上的悠轮吗 ? 在你的脑海里, 转转圆圈, 让你在边缘看它; 想象你身高8英尺, 而悠轮的圆圈正好在眼睛水平上。 该物体会以与 SHM 中大规模移动相同的方式向后移动。 它会从远处向右移动, 直到你停止旋转悠轮。 另一个例子是想象一个灯泡在晚上骑着旋转的旋转木马。 您坐在椅子上, 距离旋转木马一定的距离上, 使你能看到的系统的唯一部分是灯泡。 光的移动在你们看来会以简单的协调动作来回走。 圆形运动和简单的调调运动有很多共同点 。

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The greatest displacement of the mass from the equilibrium position is called the amplitude of the motion. One cycle refers to the complete to-and-fro motion that starts at some position, goes all the way to one side, then all the way to the other side, and returns to the original position. The period , $\begin{array}{r}T\end{array}$ , is the time required for one cycle and the frequency , $\begin{array}{r}f\end{array}$ , is the number of cycles that occur in exactly 1.00 second. The frequency, in this case, is the reciprocal of the period.
::从平衡位置上的最大质量偏移量称为运动的振幅。一个周期是指从某个位置开始、一直到一边、然后一直到另一边、然后返回到原来的位置的完全到反向运动运动。T周期是一个周期所需要的时间,频率是1.00秒后发生的周期数。在此情况下,频率是该周期的对等。

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::f=1吨

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Examples   
::实例

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When a 500. kg crate of cargo is placed in the bed of a pickup truck, the truck’s compress 4.00 cm. Assume the springs act as a single spring.
::当一箱500公斤重的货物被放在一辆小卡车的床上时,卡车的压缩器为4厘米。

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Example 1
::例1

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What is the spring constant for the truck springs?
::卡车泉泉的春季常数是多少?

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  $\begin{array}{r}k=\frac{F}{x}=\frac{(500.0\text{kg})(9.80{\text{m/s}}^2)}{0.0400\text{m}}=1.23\times {10}^5\text{N/m}\end{array}$
::k=Fx=(500.0公斤)(9.80 m/s2)0.0400 m=1.23×105 N/m

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Example 2
::例2

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How far will the springs compress if 800. kg of cargo is placed in the truck bed?
::如果800公斤货物被放在卡车床上,泉水会压缩多远?

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$\begin{array}{r}x=\frac{F}{k}=\frac{(800.0\text{kg})(9.80{\text{m/s}}^2)}{1.23\times {10}^5\text{N/m}}=0.064\text{m}=6.4\text{cm}\end{array}$
::x=Fk=(800.0公斤)(9.80米/秒)1.23×105 N/m=0.064米=6.4厘米

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Explore how the mass of the box (m) affects the distance the spring stretches (x) using the PLIX Interactive below:
::探索框( m) 的质量如何影响弹簧伸展(x) 的距离, 使用下面的 PLIX 互动 :

 

 

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Further Reading
::继续阅读

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Summary
::摘要

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• Simple harmonic motion occurs in many situations, including an object of the end of a spring, a tuning fork, a pendulum, and strings on a guitar or piano.
  ::在许多情况下会发生简单的调和运动,包括弹簧末的一个物体、一个调制叉、一个钟摆和吉他或钢琴的弦。
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• A mass oscillating on a horizontal spring is often used to analyze SHM.
  ::水平弹簧上的质量振荡经常被用来分析高温高温。
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• The restoring force for a mass oscillating on a horizontal spring is related to the displacement of the mass from its equilibrium position, $\begin{array}{r}F=-kx\end{array}$ .
  ::横向弹簧上大规模振荡的恢复力与质量从其平衡位置Fkx移位有关。
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• SHM is related to uniform circular motion when the uniform circular motion is viewed in one dimension.
  ::当从一个方面看待统一通知动议时,特别机制与统一通知动议有关。

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Review
::回顾

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1. In simple harmonic motion, when the speed of the object is maximum, the acceleration is zero.
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   1. True
      ::真实
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   2. False
      ::假假
   
   ::在简单的调和动作中,当对象的最大速度为最大时,加速度为零。
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2. In SHM, maximum displacement of the mass means maximum acceleration.
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   1. True
      ::真实
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   2. False
      ::假假
   
   ::在南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-南锥体-
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3. If a spring has a spring constant of 1.00 × 10 ³ N/m, what is the restoring force when the mass has been displaced 20.0 cm?
   ::如果春天的春季常数为1美元×103牛顿/米,那么当质量被迁移到20.0厘米时,恢复力是什么?

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Explore More 
::探索更多

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Use this resource to answer the questions that follow.
::使用此资源回答下面的问题 。

 

 

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1. What is the graph produced by a swinging pendulum's motion graphed over time?
   ::一个摇摆的钟摆运动所制作的图表是多少?
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2. How does the Exploratorium demonstrate the relationship between simple harmonic motion and circular motion? Is it convincing?
   ::探索者如何展示简单的声波运动和循环运动之间的关系?
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3. Why don't the pendulums all swing at the same rate?
   ::为什么钟摆不都按同样的速度摇摆?