Question

If \( E, L, M \) and \( G \) denote the quantities as energy, angular momentum, mass and constant of gravitation respectively, then the dimensions of \( EL^2 M^{-5} G^{-2} \) are:

• A. \( [M^0 L^1 T^0] \)
• B. \( [M^{-1} L^{-1} T^2] \)
• C. \( [M^1 L^1 T^{-2}] \)
• D. \( [M^0 L^0 T^0] \)

Hint:

This specific combination of variables often appears in orbital mechanics problems related to eccentricity or energy of orbits. Whenever you see a complex mix of \( G, M, L, \) and \( E \), check if it simplifies to a dimensionless constant.

Answer 1

The Correct Option is D

1. First, let's list the dimensional formulas of each physical quantity given in the question:
   • Energy (\( E \)): The dimensional formula for energy is \([M^1L^2T^{-2}]\).
   • Angular Momentum (\( L \)): The dimensional formula for angular momentum is \([M^1L^2T^{-1}]\).
   • Mass (\( M \)): The dimensional formula for mass is simply \([M^1]\).
   • Gravitational Constant (\( G \)): The dimensional formula for gravitational constant is \([M^{-1}L^3T^{-2}]\).
2. Next, we calculate the dimensional formula for the given expression \( E L^2 M^{-5} G^{-2} \):
   • The given expression requires:
     • \( E \) has dimension \([M^1L^2T^{-2}]\).
     • \( L^2 \) has dimension \(([M^1L^2T^{-1}])^2 = [M^2L^4T^{-2}]\).
     • \( M^{-5} \) has dimension \([M^{-5}]\).
     • \( G^{-2} \) has dimension \(([M^{-1}L^3T^{-2}])^{-2} = [M^2L^{-6}T^4]\).
   • Combine these dimensions:
     • Combine \( E \), \( L^2 \), \( M^{-5} \), and \( G^{-2} \):
       • \( [M^1L^2T^{-2}] \times [M^2L^4T^{-2}] \times [M^{-5}] \times [M^2L^{-6}T^4] \)
       • This simplifies to \([M^{1+2-5+2}L^{2+4-6}T^{-2-2+4}]\).
       • The final expression simplifies to \([M^{0}L^{0}T^{0}]\).
3. The dimensions of \( EL^2 M^{-5} G^{-2} \) are \([M^0L^0T^0]\), which represents a dimensionless quantity.
4. Therefore, the correct answer is \([M^0L^0T^0]\).