我们如何从重力势能单位中恢复力量单位?

eurocoder 06/24/2018. 4 answers, 1.182 views
forces potential potential-energy dimensional-analysis si-units

引力势$ G_ \ text {pot} $具有每单位质量的能量单位:

$$ \ bigg [\ rm \ frac {J} {kg} \ bigg] = \ bigg [\ rm \ frac {kg \ cdot m ^ 2} {s ^ 2 \ cdot kg} \ bigg] = \ bigg [\ rm \ frac {m ^ 2} {s ^ 2} \ bigg]。 $$

引力是$ F = - \ nabla G_ \ text {pot} $所以这会让我相信单位,由于渐变,我们有类似的表达式,除了额外的$ \ rm分母中的m $:

$$ \ bigg [\ rm \ frac {J} {kg \ cdot m} \ bigg] = \ bigg [\ rm \ frac {kg \ cdot m} {s ^ 2 \ cdot kg} \ bigg] = \ bigg [ \ rm \ frac {m} {s ^ 2} \ bigg]。 $$

但是力有牛顿单位:

$$ \ bigg [\ rm N \ bigg] = \ bigg [\ rm \ frac {kg \ cdot m} {s ^ 2} \ bigg] \ neq \ bigg [\ rm \ frac {m} {s ^ 2} \ bigg] $$

那么,当我采用引力势的梯度时,为什么我在单位中错过$ \ rm kg $?

4 Answers


Steeven 06/24/2018.

你使用的是错误的关系。 正确的关系不是

  • 力等于引力势的负梯度 ”但是

  • 力等于引力势能的负梯度 ”:

$$ F = - \ nabla U = - \ frac {dU} {dx} $$

这里的$ U $是potential energy ,而非potentialpotential相当于potential energy per masspotential energy per mass

米} {S ^ 2}}] = [\ mathrm {N}] $。 米} {S ^ 2} /千克}] = [\ mathrm {\压裂{米} {S ^ 2}}] = [\ mathrm {\压裂{N} {公斤}}] $。

这就是为什么(由于“每质量”特征)你在派生单位中缺少一个$ \ mathrm {kg} $。


John Rennie 06/24/2018.

我发现记住单位尺寸的最简单方法是从第二定律开始:

$$ F = ma $$

然后工作(这是一种能量形式)是力的倍数距离。

加速度单位是m / sec $ ^ 2 $,因此$ LT ^ { - 2} $。 乘以质量给出了力$ MLT ^ { - 2} $的维度,然后乘以距离给出了能量$ ML ^ 2T ^ { - 2} $的维数。

当你采用势能的梯度$ dU / dx $时,你实际上除以$ L $,所以你得到$ MLT ^ { - 2} $。 那些当然是力量的维度。


my2cts 06/24/2018.

你从引力势开始,这是每单位质量的潜在能量。 因此,您获得单位质量的力,在加速时,单位为$ m / s ^ 2 $。

顺便说一句,mksi力量单位是牛顿($ kgm / s ^ 2 $)。


abu_bua 06/24/2018.

gravitational force由下式给出

$$ \ mathbf {F} = - m \ cdot \ nabla G _ {\ text {pot}} $$

势的负梯度等于Field Strength force作用于massforce等于$ - m \ cdot \ nabla G _ {\ text {pot}} $。

电力的另一个例子,其中$ V $是电位:$ \ mathbf {F} = q \ cdot \ mathbf {E} = - q \ cdot \ nabla V $

因此,您的$ 2 ^ {nd} $等式应该更正为

$$ \ Bigg [\ cfrac {kg \ cdot m} {s ^ 2} \ Bigg] = \ Bigg [kg \ cdot \ cfrac {J} {kg \ cdot m} \ Bigg] = \ Bigg [kg \ cdot \ cfrac {kg \ cdot \ frac {m ^ 2} {s ^ 2}} {kg \ cdot m} \ Bigg] $$

$$ \ Bigg [\ cfrac {kg \ cdot m} {s ^ 2} \ Bigg] = \ Bigg [\ cfrac {kg \ cdot m} {s ^ 2} \ Bigg] $$


知道Joule给出的potential $ \ mathbf U $,让你计算一下力

$$ \ mathbf F = - \ nabla U \ quad。$$


btw:将万有引力视为经典力量是错误的 - >参见Einstein's Relativity Theory


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