Question

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 \(\frac{M}{m}\) for which, the gravitational potential energy of the system becomes maximum is x : 1. The value of x is ________. 

 

Hint:

To maximize a negative function, you need to minimize its absolute value. The standard calculus approach of setting the first derivative to zero works perfectly. Be careful to sum the potential energy for all pairs in the system.

Answer 1

Correct Answer: 2

Step 1: Understanding the Question:
We have four masses arranged at the corners of a square. The total gravitational potential energy (GPE) of this system depends on the value of 'm'. We need to find the ratio M/m for which this GPE is maximum.
Step 2: Key Formula or Approach:
The gravitational potential energy between two masses \(m_1\) and \(m_2\) separated by a distance \(r\) is \(U = -G\frac{m_1m_2}{r}\).
The total GPE of the system is the sum of the potential energies of all possible pairs of masses. In a square with four masses, there are 6 pairs (4 sides and 2 diagonals). To find the maximum GPE, we will differentiate the total potential energy expression with respect to 'm' and set the derivative to zero.
Step 3: Detailed Explanation:
The arrangement has masses 'm' and 'M-m' at adjacent corners. The four masses at the vertices are \(m_1 = m\), \(m_2 = M-m\), \(m_3 = m\), \(m_4 = M-m\). The side length of the square is 'd', and the diagonal length is \(d\sqrt{2}\).
Let's calculate the total GPE (\(U_{total}\)) by summing the energy of all 6 pairs:
4 pairs along the sides (distance d): \(U_{sides} = -G\frac{m(M-m)}{d} -G\frac{(M-m)m}{d} -G\frac{m(M-m)}{d} -G\frac{(M-m)m}{d} = -4G\frac{m(M-m)}{d}\)
2 pairs along the diagonals (distance \(d\sqrt{2}\)): \(U_{diag} = -G\frac{m \cdot m}{d\sqrt{2}} -G\frac{(M-m)(M-m)}{d\sqrt{2}} = -\frac{G}{d\sqrt{2}}(m^2 + (M-m)^2)\)
Total GPE: \[ U_{total} = -\frac{G}{d} \left[ 4m(M-m) + \frac{1}{\sqrt{2}}(m^2 + (M-m)^2) \right] \] To maximize \(U_{total}\), which is a negative quantity, we must minimize its magnitude. Let the term in the brackets be \(f(m)\). \[ f(m) = 4mM - 4m^2 + \frac{1}{\sqrt{2}}(m^2 + M^2 - 2mM + m^2) = 4mM - 4m^2 + \frac{1}{\sqrt{2}}(2m^2 - 2mM + M^2) \] For maximum U, we need \(\frac{dU_{total}}{dm} = 0\), which means \(\frac{df(m)}{dm} = 0\). \[ \frac{df}{dm} = 4M - 8m + \frac{1}{\sqrt{2}}(4m - 2M) = 0 \] \[ 4M - 8m + \frac{4}{\sqrt{2}}m - \frac{2}{\sqrt{2}}M = 0 \] \[ 4M - 8m + 2\sqrt{2}m - \sqrt{2}M = 0 \] Group terms with M and m: \[ M(4 - \sqrt{2}) = m(8 - 2\sqrt{2}) \] \[ M(4 - \sqrt{2}) = m \cdot 2(4 - \sqrt{2}) \] Cancel the \((4 - \sqrt{2})\) term from both sides: \[ M = 2m \] \[ \frac{M}{m} = 2 \] The ratio is given as x:1. \[ \frac{M}{m} = \frac{x}{1} \implies x = 2 \] Step 4: Final Answer:
The value of x is 2.