Question

SI unit of gravitational constant (G) is :

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

Hint:

To find the units of a physical constant from a formula:
1. Write down the formula involving the constant. 2. Rearrange the formula to make the constant the subject. 3. Substitute the SI units of all other quantities in the formula. 4. Simplify the resulting expression of units. This method is widely applicable for deriving units of various physical constants (e.g., Planck's constant, specific heat capacity, etc.).

Answer 1

The Correct Option is D

Concept: The gravitational constant (\(G\)) appears in Newton's Law of Universal Gravitation. We can derive its SI unit by rearranging this law. Step 1: Recall Newton's Law of Universal Gravitation The formula for the gravitational force (\(F\)) between two masses (\(m_1\) and \(m_2\)) separated by a distance (\(r\)) is: \[ F = G \frac{m_1 m_2}{r^2} \] where \(G\) is the universal gravitational constant. Step 2: Rearrange the formula to solve for G To find the units of \(G\), we first need to isolate \(G\) on one side of the equation: Multiply both sides by \(r^2\): \[ F r^2 = G m_1 m_2 \] Now, divide both sides by \(m_1 m_2\): \[ G = \frac{F r^2}{m_1 m_2} \] Step 3: Determine the SI units for each quantity in the rearranged formula
\(F\) (Force): The SI unit is Newton (N).
\(r\) (distance): The SI unit is meter (m). So, \(r^2\) will have units of \(\text{m}^2\).
\(m_1\) (mass): The SI unit is kilogram (kg).
\(m_2\) (mass): The SI unit is kilogram (kg).
Therefore, \(m_1 m_2\) will have units of \(\text{kg} \times \text{kg} = \text{kg}^2\). Step 4: Substitute the units into the expression for G \[ \text{Units of G} = \frac{(\text{Units of } F) \times (\text{Units of } r^2)}{(\text{Units of } m_1 m_2)} \] \[ \text{Units of G} = \frac{\text{N} \times \text{m}^2}{\text{kg}^2} \] This can also be written as: \[ \text{Units of G} = \text{Nm}^2\text{kg}^{-2} \] or \[ \text{Units of G} = \text{N} \cdot \text{m}^2 / \text{kg}^2 \] Comparing this with the given options, option (4) \(\text{Nm}^2 \text{kg}^{-2}\) matches our derived unit. Option (1) \(\text{Nm}^2 \text{kg}^{-1}\) is incorrect because the denominator should be \(\text{kg}^2\). Therefore, the SI unit of the gravitational constant (G) is \(\text{Nm}^2 \text{kg}^{-2}\).