首先是一个小视频,从1e+24到1e-16,

测量,误差精度很重要!误差永远存在!

假设动物的尺寸是S,质量是M,骨骼长度为L,骨骼的横截面是A

{l \propto s}

{m \propto {s^3} \propto {l^3}}

{preause \propto \frac{{weight}}{A} \propto \frac{m}{{{d^2}}}}

Now follow me closely.

If the pressure is higher than a certain level the bones will break.

Therefore, for an animal not to break its bones when the mass goes up by a certain factor let's say a factor of four in order for the bones not to break d squared must also go up by a factor of four.

That's a key argument in the scaling here.

You really have to think that through carefully.

Therefore, I would argue that the mass must be proportional to d squared.

This is the breaking argument.

Now compare these two.

{m \propto {d^2}}

{{d^2} \propto {l^3} \Rightarrow d \propto {l^{\frac{3}{2}}}}

这里就是推导出来的长度以及骨骼的厚度的关系了(尽管事实上是不正确的。。。)

然后是单位的匹配。纲量分析。

假设苹果质量m,从高度h落下,时间为t,

{t \propto {h^\alpha }{m^\beta }{g^\gamma }}

{{{[T]}^1} = {{[L]}^\alpha }{{[M]}^\beta }\frac{{{{[L]}^\gamma }}}{{{{[T]}^{2\gamma }}}}}

{\beta = 0}

{\alpha + \gamma = 0}

{1 = - 2\gamma }

{t = C\sqrt {\frac{h}{g}}\propto\sqrt h }

不确定度的计算