Understanding the Universal Constant of Gravitation

The question asks to identify the value of the Universal Constant of Gravitation. This constant is a fundamental physical constant used in Newton's Law of Universal Gravitation.

Newton's Law of Universal Gravitation states that every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, this force ($F$) between two objects with masses $m_1$ and $m_2$ separated by a distance $r$ is given by the formula:

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

In this formula, $G$ is the Universal Constant of Gravitation. It is called 'universal' because its value is believed to be the same throughout the entire universe, regardless of the nature of the masses or the medium between them.

Value and Units of the Universal Gravitation Constant

The value of the Universal Constant of Gravitation ($G$) has been determined experimentally. The accepted value is approximately:

$$G \approx 6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2$$

Let's look at the units. From the formula $F = G \frac{m_1 m_2}{r^2}$, we can rearrange to find the units of $G$:

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

The units are: (Units of Force) × (Units of Distance)$^2$ / (Units of Mass)$^2$.

In SI units:

• Force is measured in Newtons (N).
• Distance is measured in meters (m).
• Mass is measured in kilograms (kg).

So, the units of $G$ are $\text{N} \cdot \text{m}^2 / \text{kg}^2$, which is written as $\text{Nm}^2/\text{kg}^2$. This matches the units given in the options.

Analyzing the Options

Now let's compare the given options with the known value of the Universal Constant of Gravitation:

• Option 1: $9.8 \text{ Nm}^2/\text{kg}^2$. This value is close to the acceleration due to gravity on Earth ($g \approx 9.8 \text{ m/s}^2$), not the Universal Constant of Gravitation ($G$). Also, the units are incorrect for $G$.
• Option 2: $6.76 \times 10^{-10} \text{ Nm}^2/\text{kg}^2$. This value is close to the magnitude but the exponent is $-10$, not $-11$.
• Option 3: $6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2$. This matches the commonly accepted value and units for the Universal Constant of Gravitation.
• Option 4: $6.67 \times 10^{10} \text{ Nm}^2/\text{kg}^2$. This value has a positive exponent of $10$, making it a very large number, vastly different from the actual very small value of $G$.

Based on the comparison, Option 3 correctly represents the value and units of the Universal Constant of Gravitation.

Conclusion on Universal Gravitation Constant

The Universal Constant of Gravitation, $G$, is a crucial constant in physics that quantifies the strength of the gravitational force between masses. Its experimentally determined value is approximately $6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2$. Among the given options, only option 3 provides this correct value and the appropriate units.

Revision Table: Universal Constant of Gravitation Key Facts

Additional Information on Universal Gravitation Constant

While $g$ (acceleration due to gravity) varies depending on location (height, latitude, mass of the celestial body), the Universal Constant of Gravitation, $G$, is a fundamental constant of nature that is expected to be the same throughout the universe. The value of $G$ is incredibly small, which explains why gravitational force is significant only when at least one of the interacting masses is very large (like a planet or star). Measuring $G$ accurately is quite challenging due to the weakness of the gravitational force between laboratory-sized objects. Early measurements were done by scientists like Henry Cavendish.