Question

The dimensional formula of the gravitational constant is \( M^a L^b T^c \), the values of \( a \), \( b \), and \( c \) are respectively:

• A. \( 1, 3, -2 \)
• B. \( -1, 3, 2 \)
• C. \( -1, 3, -2 \)
• D. \( 1, -3, 2 \)
• E. \( 1, -3, -2 \)

Hint:

To find the dimensional formula of physical constants, use the fundamental equations where the constant is present, and balance the dimensions of all quantities involved.

Answer 1

The Correct Option is C

The gravitational constant \( G \) appears in Newton's law of gravitation: \[ F = \frac{G m_1 m_2}{r^2} \] where: - \( F \) is the force with dimensional formula \( [M L T^{-2}] \)
- \( m_1 \) and \( m_2 \) are masses with dimensional formula \( [M] \)
- \( r \) is the distance with dimensional formula \( [L] \)
The dimensional formula of \( G \) can be derived as follows: \[ [F] = \frac{[G] [M]^2}{[L]^2} \] Substituting the dimensional formulas: \[ [M L T^{-2}] = \frac{[G] M^2}{L^2} \] Solving for \( [G] \): \[ [G] = \frac{M L^3 T^{-2}}{M^2} = M^{-1} L^3 T^{-2} \] Thus, the dimensional formula of \( G \) is: \[ [G] = M^{-1} L^3 T^{-2} \] Therefore, comparing this with \( M^a L^b T^c \), we get: \[ a = -1, \quad b = 3, \quad c = -2 \] Thus, the correct values of \( a \), \( b \), and \( c \) are \( -1, 3, -2 \). \[ \boxed{(-1, 3, -2)} \]