Question

SI unit of universal gravitational constant (G) is :

• A. \(\text{Nm}^3 \text{kg}^{-2}\)
• B. \(\text{Nm}^2 \text{kg}^{-1}\)
• C. \(\text{Nm kg}^{-2}\)
• D. \(\text{Nm}^2 \text{kg}^{-2}\)

Hint:

1. Start with Newton's Law of Gravitation: \(F = G \frac{m_1 m_2}{r^2}\). 2. Rearrange to solve for G: \(G = \frac{F r^2}{m_1 m_2}\). 3. Substitute the SI units for each term:
F (Force) \(\rightarrow\) N (Newton)
r (distance) \(\rightarrow\) m (meter), so \(r^2 \rightarrow m^2\)
\(m_1, m_2\) (mass) \(\rightarrow\) kg (kilogram), so \(m_1 m_2 \rightarrow kg^2\) 4. Unit of G = \(\frac{\text{N} \cdot \text{m}^2}{\text{kg}^2} = \text{Nm}^2\text{kg}^{-2}\).

Answer 1

The Correct Option is D

Concept: The SI unit of the universal gravitational constant (G) can be derived from Newton's Law of Universal Gravitation. Newton's Law of Universal Gravitation states: \(F = G \frac{m_1 m_2}{r^2}\), where:
\(F\) is the gravitational force between two masses.
\(G\) is the universal gravitational constant.
\(m_1\) and \(m_2\) are the two masses.
\(r\) is the distance between the centers of the two masses. Step 1: Rearrange the formula to solve for G \[ G = \frac{F r^2}{m_1 m_2} \] Step 2: Identify the SI units for each quantity in the formula
Force (\(F\)): Newton (N)
Distance (\(r\)): meter (m)
Mass (\(m_1\), \(m_2\)): kilogram (kg) Step 3: Substitute the SI units into the rearranged formula for G Unit of G = \(\frac{(\text{Unit of } F) \times (\text{Unit of } r)^2}{(\text{Unit of } m_1) \times (\text{Unit of } m_2)}\) Unit of G = \(\frac{\text{N} \times (\text{m})^2}{\text{kg} \times \text{kg}}\) Unit of G = \(\frac{\text{N m}^2}{\text{kg}^2}\) Step 4: Express the unit using negative exponents This can also be written as: Unit of G = \(\text{Nm}^2 \text{kg}^{-2}\) This matches option (4).