I draw a graph here of the amount by which I pull the spring versus the force it exerts.

我在这里画个图,纵轴是弹簧的拉伸量,横轴是力的大小

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Newton is not going to come and tell you what force the spring will exert when it's pulled by some amount.

牛顿不会过来告诉你,弹簧被拉伸一定量时会产生多大的力

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Now I can measure that, because I pull it by one centimeter and I find the acceleration it exerts on a known mass.

现在就能测量这个力了,因为把弹簧拉伸 1 厘米,我就知道它让一个质量已知的物体产生的加速度

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We want to know how much force it exerts when I pull it by a certain amount.

我们想知道当我们将弹簧拉伸一定量时,它能产生多大的力

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I tie a spring to my back and I anchor it to the wall and see how much the spring stretches before the two forces balance.

我把弹簧一端系在背后,另一端固定在墙上,观察两个力平衡时弹簧的伸长量

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First, we got to make sure the spring exerts a fixed force every time.

首先,要保证,弹簧每次产生的力是恒定的

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Namely, if you pull it by so much, that's the force it will exert.

也就是,将弹簧拉伸这么长,它产生的力就有这么大

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We don't know what the force of the spring is.

我们不知道弹簧的力多大

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A spring will, on the other hand, exert a force.

但是一个弹簧可以提供一个力

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Then, I bring the elephant and I pull the spring by the same amount and I find the acceleration of the elephant and the denominator is obviously the mass of the elephant.

然后我再找来大象,用相同的力去拉弹簧,大象就具有了加速度,很明显,分母就是大象的质量

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You cannot say to me, "Take a force, due to a spring, and see what force it applies and divide by the acceleration and get the mass, " because we haven't defined force, either.

你不能说,"给弹簧施加一个力,看看它能提供多大的力,然后除以加速度,得到质量",因为我们还没给力下一个定义

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For example, how do I know that when I pull the spring the first time for the mass, it exerted the same force when I pulled the spring the second time for the elephant?

例如 怎样才能知道,第一次连着物体拉弹簧时,产生的力和,第二次连着大象拉弹簧时产生的力是相等的呢

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This is the force of the spring.

这个是弹簧产生的力

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The force may taper off, the force may not be given by a straight line, but for modest deformations, every spring will have a linear regime in which the force is linearly proportional to the stretching.

它产生的力可能会渐渐消失,力的曲线可能不是一条直线,而有轻微的弯曲,每个弹簧都有一个线性区间,在这一区间内,力和伸长量是线性关系

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That m times a is the force the spring is exerting.

与 a 的乘积就等于弹簧产生的力

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It says, if you pull it to the right, so that x is positive, the spring will exert a force which is in the negative direction; that's why you have a minus sign.

就是说,如果你向右拉,x 取正值,弹簧产生的力方向为负,这就是这里用负号的原因

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We did the comparison with the elephant or something; a is the second derivative of x and for this problem, when F is due to a spring, we know the force is that by studying the spring.

我们也已经把它与大象或其它东西作过比较,a 是 x 的二阶导数,在这个问题中,F 是由弹簧产生的,我们在讨论弹簧问题时已经知道了力的大小

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