A body of mass 2 M splits into four masses { m , M - m , m , M - m } , which are rearranged to form a square as shown in the figure. The ratio of M m for which, the gravitational potential energy of the system becomes maximum is x   :   1 . The value of x is ________.

Question

A body of mass 2 M splits into four masses { m , M - m , m , M - m } , which are rearranged to form a square as shown in the figure. The ratio of M m for which, the gravitational potential energy of the system becomes maximum is x &#160; : &#160; 1 . The value of x is ________.

Aicharya · Accepted Answer

U = G m 2 a + G ( M - m ) 2 a + 2 G m ( M - m ) a + 2 G m ( M - m ) 2 a For maxima or minima of potential energy d U d m = 0 d U d m = G a ( 2 &#160; m - 2 ( M - m ) + ( M - 2 &#160; m ) 2 + 2 ( M - 2 &#160; m ) ( 4 m - 4 m - 2 2 m ) + ( - 2 M + 2 M + 2 M ) = 0 ; &#160;&#160; 2 M = 2 2 m ; &#160;&#160; M m = 2

A body of mass 2 M splits into four masses { m , M - m , m , M - m } , which are rearranged to form a square…

Solution