Keying  Guan

Department of Mathematics, School of Science, Beijing Jiaotong University

Email: keying.guan@gmail.com

Just as Einstein demonstrated that mass equivalent to energy, force is equivalent to power. This is implicit in modern analytical mechanics and revealed in the author's research blog post 

没有不做功的力--马德堡半球实验、举重运动消耗的卡路里及辐射压公式 2022-07-01

This equivalence can be expressed precisely as that every newton of force acting on 1 kg of material is equivalent to outputting 1 watt of power. 

Importantly, this relationship exists instantaneously and does not depend on whether the force remains constant over time, nor on the mass and instantaneous velocity of the object being acted upon. Therefore, the equivalence relationship between force and power is an objective and natural law.

This law is not only reflected in the Magdeburg hemisphere experiment, but also in the heat consumed by weightlifters, and in the plastic deformation, heat generation, fatigue and fracture of the extruded (or stretched) material per unit time degree, etc.

In that paper, the authors emphasized that it is easy to overlook the equivalence of force and power when dealing with "static forces" related problems. Ignoring this rule can lead to miscalculation of the actual problem.

For example, many people mistakenly believe that a person who is standing and carrying a weight is not doing work when he is not pushing the weight higher further, etc., so he consumes the same amount of physical energy as if he were not carrying the weight. 

Such misjudgments are more seriously reflected in the Maxwell-Bartoli radiation pressure formula (including Krönig's pressure formula for ideal gas) with reflection coefficients in modern physical science theories. In fact, these pressure formulations only use the law of conservation of momentum and ignore the force and power equivalence associated with the law of conservation of energy.  So, the Maxwell-Bartoli formula erroneously predicts that the radiation pressure on the surface of the strongly reflective material with a small ability to receive radiation is greater than the radiation pressure on the surface of the weakly reflective material with a large ability to receive radiation. This prediction both contradicts the experimental facts of the Crookes radiometer and grossly underestimates Crookes' scientific contributions. 

To answer these questions, the authors first looked at the number of calories weightlifters burn during a single weightlifting session. The specific question is: If a 100kg barbell is forced off the ground within 3 seconds, lifted to a 2-meter position (snatch, clean and jerk are not considered here), and then held in this position for 3 seconds, how many calories does the athlete need to burn ?

The author has never seen a rigorous answer to this question in an academic article. And it is reported that there is considerable debate on the issue.

If you only lift the barbell from the ground to a height of 2 meters according to the usual physical knowledge, the gravitational potential energy of the barbell is equivalent to an increase of 468.6 calories (note: because this is a simple mechanical calculation, this calculation is omitted). If the answer to the above question is this, it is obviously wronged to the lifter.

In fact, based on the force-power relationship, an athlete needs to burn 234.3 calories (or 980 joules) of energy per second just to support a 100-kilogram (9,800 Newton) barbell without hitting the ground for 6 seconds. Additionally, raising the bar to a height of 2 meters in those 6 seconds requires the athlete to burn an additional 468.6 calories. Therefore,  the athlete burns 1874.4 (= 6 x 234.3 + 468.6) calories during this 6 second movement.

The above calculation is purely mechanical. The same amount of energy is consumed if this action is performed using a machine.  

This example shows that just as weightlifters are wronged, an inanimate column is wronged if it is thought that using an inanimate column to support a weight will not consume energy as if it were not supporting that weight.

Regarding the wrong Maxwell-Bartoli formula and ideal air pressure formula mentioned, the author also deduces a more reasonable formula based on the revealed force-power equivalence and the exact solutions of the elastic collision of two particles (note: the exact solutions can be find in the blog post Relativistic elastic collision of two particles in 2D,  and its revised version is attached in the above mentioned blog post,

where <S_incident> is the power of the incident flow through a unit area, R_reflection is the reflection coefficient of the illuminated object, s is the unit time (1 second), and m is the unit length (1 meter). 

This formula is qualitatively consistent with the experiments of the Crooles radiometer

Since the above results are new and believed to be related to the basis of physics, the author sincerely hopes that relevant scholars will pay attention to and study the above mentioned blog post. Criticism and suggestions welcome.

The Chinese version

中文译文：                                                  力等价于功率

正如爱因斯坦证明质量等同于能量一样，力等价于功率。这个等价关系隐含在现代分析力学中，并在作者的如下研究性博客文章中揭示出来

没有不做功的力--马德堡半球实验、举重运动消耗的卡路里及辐射压公式 2022-07-01,

当使用 SI 系统时，这种等价关系可以精确地表示为，作用在 1 千克的物质上的每牛顿力相当于输出 1 瓦特的功率。

重要的是，这种关系即时地存在，不依赖于力是否在一段时间内保持恒定，也不依赖于被作用物体的质量和瞬时速度。 因此，力与功率的等价关系是客观的、自然的规律。

这一规律不仅体现在马格德堡半球实验中，还体现在举重运动员所消耗的热量，以及单位时间度内挤压（或拉伸）材料的塑性变形、发热、疲劳和断裂等方面。

在那篇论文中，作者强调，在处理“静态力”相关问题时，很容易忽略力和功率的等价性。 忽略这条规则可能会导致对实际问题的误判。

例如，很多人错误地认为，一个站着负重的人在没有把重物推得更高的时候就不是在做功等，所以他消耗的体能与没有负重一样多。

这种误判在现代物理科学理论中带有反射系数的麦克斯韦-巴托利辐射压力公式（包括克朗尼希理想气体压力公式）中体现得更为严重。 事实上，这些压力公式只使用了动量守恒定律，而忽略了与能量守恒定律相关的力和功率等价性。 因此，麦克斯韦-巴托利公式错误地预测了接收辐射能力小的强反射材料表面的辐射压力大于接收辐射能力大的弱反射材料表面的辐射压力。 这一预测既与克鲁克斯辐射计的实验事实相矛盾，又严重低估了克鲁克斯的科学贡献。

为了回答这些问题，作者首先研究了举重运动员在单次举重训练中燃烧的卡路里数量。 具体问题是：如果一个100kg的杠铃在3秒内用力离地，举到2米的位置（抓举，挺举，这里不考虑），然后在这个位置保持3秒，为此运动员需要燃烧多少卡路里？

作者从未在学术文章中看到过对这个问题的严谨回答。 据报道，在这个问题上存在相当大的争论。

如果按照通常的物理知识，只将杠铃从地面举到2米的高度，杠铃的重力势能相当于增加了468.6卡路里（注意：因为这是一个简单的机械计算，所以这个计算 被省略）。 如果上述问题的答案即是这样的，那显然是委屈了举重者。

事实上，根据力量-力量的关系，一名运动员每秒需要燃烧 234.3 卡路里（或 980 焦耳）的能量才能支撑 100 公斤（9,800 牛顿）的杠铃而不触地 6 秒。 此外，在这 6 秒内将杠铃提升到 2 米的高度需要运动员额外燃烧 468.6 卡路里的热量。 因此，运动员在这 6 秒的运动中燃烧了 1874.4 (= 6 x 234.3 + 468.6) 卡路里。

以上计算是纯力学的。如果使用机器完成此动作也会消耗同样的能。

这个例子表明，就像举重运动员被冤枉一样，如果认为使用无生命的柱子支撑重量不会消耗能量，就像它没有支撑那个重量一样，那么无生命的柱子同样地被冤枉了。

对于错误的 Maxwell-Bartoli 公式和理想气压公式，作者还根据揭示的力-功率等价性和两个粒子弹性碰撞的精确解推导出了一个更合理的公式（注：精确解可以在 博客文章 2D 中两个粒子的相对论弹性碰撞，其修订版附在上述的博文上）

其中 <S_incident> 是入射流通过单位面积的功率，R_reflection，是被照物体的反射系数系数，s 是单位时间 (1 秒)，m 是单位长度(1 米)。

此公式定性上与 Crooles 辐射计的实验一致。

由于上述结果是新的，相信与物理学基础有关，作者衷心希望相关学者关注和研究上述博文. 欢迎批评和建议。