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利用圆形轨道的稳定性规律,推导出了地球被杠杆撬起高度△r后在新轨道上所受合力表达式为F(r_0+△r)=-2GMm/r_0~3·△r=-k·△r.从这一表达式可以看出,地球撬起后的运动轨迹与弹簧振子的运动轨迹类似,即虽然地球运动的轨道半径由r_0变为r_0+△r,但随着时间的变化r_0+△r始终保持在r_0附近作微小简谐振动,不会远离原轨道;同时,根据k值(5×10~(11)),估算出了阿基米德需要持续约33年时间才能将地球被撬起仅仅1cm.从而,有效证明了阿基米德利用杠杆将地球撬起的设想是不能够实现.
Using the law of stability of circular orbit, we deduce that the resultant force on the new orbit when the earth is levered by the lever △ r is F (r_0 + Δr) = - 2GMm / r_0 ~ 3 · Δr = -k · △ r. From this expression, we can see that the earth’s trajectory after prying is similar to that of the spring oscillator. That is, although the orbital radius of earth motion changes from r_0 to r_0 + Δr, with the change of time r_0 + △ r is always kept in the vicinity of r_0 as a simple harmonic vibration, not far away from the original orbit; the same time, based on the k value (5 × 10 ~ (11)), it is estimated that Archimedes need to last about 33 years to earth Was prized up to only 1 cm. Thus, it was effectively proved that Archimedes’ use of leverage to lift the earth would not be realized.