question_answer 39)

                  Supposing Newton?s law of gravitation for gravitation force \[{{F}_{1}}\] and \[{{F}_{2}}\] between two masses \[{{m}_{1}}\] and \[{{m}_{2}}\] at positions \[{{r}_{1}}\] and \[{{r}_{2}}\] read \[{{F}_{1}}=-{{F}_{2}}=\frac{{{{\vec{r}}}_{12}}}{r_{12}^{3}}\,\,GM_{0}^{2}\,\,{{\left( \frac{{{m}_{1}}{{m}_{2}}}{M_{0}^{2}} \right)}^{n}}\] where \[{{M}_{o}}\] is a constant of dimension of mass, \[{{r}_{12}}={{r}_{1}}-{{r}_{2}}\] and N is a number. In such a case.                 (a) the acceleration due to gravity on earth will be different for different objects.                 (b) none of the three laws of Kepler will be valid.                 (c) only the third law will become invalid.                 (d) for n negative, an object lighter than water will sink in water.

Answer:

                  (a, c, d) \[|\vec{F}|\,\,=\frac{GM_{0}^{2}}{r_{12}^{2}}\,\,{{\left( \frac{{{m}_{1}}{{m}_{2}}}{M_{0}^{2}} \right)}^{n}}\]                 \[=\frac{GM_{0}^{2\,(1-n)}}{r_{12}^{2}}\,\,{{({{m}_{1}}{{m}_{2}})}^{n}}\]                 \[g=\,\frac{|\vec{F}|}{mass}\]