A body of mass (2M) 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 ............ .

Solution

Physics sir · 4 steps

Write U as a function of m alone, differentiate with respect to m, and the turning point gives M = 2m.

1. Count the pairs. Four masses make six pairs. Each of the 4 sides joins an m to an M-m at distance d. The two diagonals, of length √2 d, join m to m and (M-m) to (M-m).

   U = -G/d[4m(M-m) + (m² + (M-m)²)/√2]

2. Differentiate in m. M is fixed, so only m varies. The derivative of the bracket is 4(M-2m) + (2m - 2(M-m))/√2 = 4(M-2m) - √2(M-2m) = (4-√2)(M-2m).
3. Find the extremum. Setting dU/dm = 0 gives M - 2m = 0, so m = M/2. This is the only turning point of U for 0 < m < M. It is the extremum the question asks for, and the size of U is largest there.
4. So. M/m = 2, so the ratio is 2 : 1 and x = 2.

The common slip: 0.5. Reporting m/M instead of M/m gives 0.5, not 2. Read which ratio the question asks for.

The idea to keep. When the masses are split by one variable, write the total potential energy as a function of that variable. Then set its derivative to zero. Count every pair once, sides and diagonals.

Answer from the NTA final answer key. Solution by PrepWiser.