Question

The dimensions of universal gravitational constant are 

• A. ML² T^–1
• B. M^–2L³ T^–2
• C. M^–2L² T^–1
• D. M^–1L³ T^–2

Answer 1

The Correct Option is D

The universal gravitational constant, denoted as G, is used in Newton's law of universal gravitation. According to the law, the force F of attraction between two masses m_1 and m_2 separated by a distance r is given by:

F = \frac{G \cdot m_1 \cdot m_2}{r^2}

Where:

• F is the gravitational force with dimensions [MLT^{-2}] (force)
• m_1 and m_2 are masses with dimensions [M]
• r is the distance with dimension [L]

Hence, rearranging for G, we get:

G = \frac{F \cdot r^2}{m_1 \cdot m_2}

Substitute the dimensions of each term:

• F = [MLT^{-2}]
• r^2 = [L^2]
• m_1 \cdot m_2 = [M] \cdot [M] = [M^2]

Thus, the dimensions of G will be:

G = \frac{[MLT^{-2}] \cdot [L^2]}{[M^2]} = [M^{-1}L^3T^{-2}]

This matches the correct answer, which is:

Correct Answer: M^{-1}L^3T^{-2}

This dimensional formula is crucial in understanding how the gravitational constant balances the units in gravitational force calculations. It demonstrates the relationship between force, distance, and mass in the context of gravitational interactions.