- Newton's law of universal gravitation is given by \(F = G \frac{m_1 m_2}{r^2}\).
- Rearranging the equation gives \(G = \frac{F r^2}{m_1 m_2}\), which results in the SI units \(\text{N} \cdot \text{m}^2/\text{kg}^2\).
- The experimentally measured value for the gravitational constant is approximately \(6.67 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2\).
- Comparing this to the provided options, the correct choice is \(\boxed{6.67 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2}\).
Explanation
- Deriving the Units of the Gravitational Constant
The gravitational constant, denoted as \(G\), is a fundamental physical constant that appears in Newton's law of universal gravitation. This law describes the gravitational force \(F\) acting between two point masses, \(m_1\) and \(m_2\), separated by a distance \(r\), and is expressed by the formula:
\[F = G \frac{m_1 m_2}{r^2}\]
To determine the correct value and units, we first need to isolate \(G\) in the equation. By rearranging the formula, we get:
\[G = \frac{F r^2}{m_1 m_2}\]
Next, we analyze the dimensions and units in the International System of Units (SI). Force \(F\) is measured in Newtons (\(\text{N}\)), distance \(r\) is measured in meters (\(\text{m}\)), and mass \(m\) is measured in kilograms (\(\text{kg}\)). Thus, substituting these units into the rearranged formula, the unit of \(G\) is:
\[\text{Units of } G = \frac{\text{N} \cdot \text{m}^2}{\text{kg}^2}\]
This unit identifies the correct physical dimension for the gravitational constant.
- Identifying the Correct Value
The numerical value of the gravitational constant \(G\) has been experimentally determined through precise laboratory measurements. Its accepted value is approximately:
\[G \approx 6.674 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2\]
Comparing this result with the options provided:
- \(6.67 \times 10^{-11} \text{ m/s}\) (Incorrect units)
- \(9.8 \text{ m}^2/\text{s}^2 \times 10^s \text{ m/kg}\) (Incorrect value and units)
- \(9.8 \text{ N} \cdot \text{m}^2/\text{kg}^2 \times 10^8 \text{ m/s}\) (Incorrect value and units)
- \(6.67 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2\) (Correct value and units)
The only option that matches both the standard physical constant and its corresponding SI units is the final one. Thus, the value of the gravitational constant is \(\boxed{6.67 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2}\).
Examples
The gravitational constant \(G\) is crucial for calculating the forces between celestial bodies. For instance, astronomers use this constant, along with the masses of the Earth and the Moon and the distance between them, to calculate the gravitational pull that keeps the Moon in orbit around the Earth. Knowing \(G\) allows us to predict the behavior of planetary systems and understand how gravity shapes the universe at both local and cosmic scales.