question_answer

Three uniform spheres, with masses \[{{m}_{A}}=350\,kg,\] \[{{m}_{B}}=2000\,kg,\] and \[{{m}_{C}}=500\,kg\]have the (x, y) coordinates (0, 0) cm, (- 80, 0) cm and (40, 0) cm respectively. The gravitational potential energy, U, of the system and change in its value in terms of increase or decrease, if the sphere of mass  is removed, may be given as

A)  \[U=-1.92\times {{10}^{-4}}J\] and its value shall decrease if the sphere B is removed

B)  \[U=-1.92\times {{10}^{-4}}J\] and its value shall increase if the sphere B is removed

C)  \[U=-1.43\times {{10}^{-4}}J\] and its value shall decrease if mg is removed

D)  \[U=-1.43\times {{10}^{-4}}J\] and its value shall increase if \[{{m}_{B}}\] is removed

Correct Answer: D

Solution :

                 \[U=-G\left[ \frac{{{m}_{A}}{{m}_{B}}}{{{r}_{AB}}}+\frac{{{m}_{B}}{{m}_{C}}}{{{r}_{BC}}}+\frac{{{m}_{C}}{{m}_{A}}}{{{r}_{CA}}} \right]\]                 \[=6.7\times {{10}^{-11}}\]                          \[\left[ \frac{350\times 2000}{80\times {{10}^{-2}}}+\frac{2000\times 500}{120\times {{10}^{-2}}}+\frac{350\times 500}{40\times {{10}^{-2}}} \right]\]                 \[=-6.7\times {{10}^{-11}}[8750+8333.33+4375]\times {{10}^{+2}}\]                 \[=-6.7\times {{10}^{-9}}\times 21458.33\]                 \[=-143770.81\times {{10}^{-9}}=-1.43\times {{10}^{-4}}J\]  Hence gravitational potential energy\[U=-1.43\times {{10}^{-4}}J\] and its value shall increase if \[{{m}_{B}}\] removed.