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Newton's law of universal gravitation is a cornerstone principle in physics, elucidating how every point mass attracts every other point mass in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This can be mathematically expressed as:\[F = \frac{G \times m_1 \times m_2}{r^2}\]where:

• \( F \) is the gravitational force between two objects,
• \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^2 / \mathrm{kg}^2 \),
• \( m_1 \) and \( m_2 \) are the masses of the two objects in kilograms,
• \( r \) is the distance between the centers of the two masses in meters.

This law beautifully shows the universality of gravitational force, a principle that applies equally here on Earth and to celestial objects far away in space. Thus, even when objects are floating in space far from other influences, they still exert gravitational forces on each other due to their masses.

In physics, it is crucial to understand the distinction between mass and weight, particularly when converting from one to the other. Weight is a force, measured in newtons (N), that results from the gravitational pull on an object's mass. On Earth, weight can be calculated as \( W = m \times g \), where \( g \approx 9.81 \mathrm{~m/s^2} \) is the gravitational acceleration. To convert from weight to mass, which is a measure of the amount of matter in an object (measured in kilograms), you rearrange the formula:\[m = \frac{W}{g}\]In the original exercise, each cannonball had a weight of \(4.00\) kN, equivalent to \(4.00 \times 10^3\) N. By applying the conversion formula, their mass is found to be approximately \(407.75\) kg per cannonball. This conversion is crucial for calculating gravitational force, which depends directly on the masses of the interacting bodies. Converting weight to mass enables accurate application of Newton's gravitational formula outside the preferred context of Earth's gravitational influence.

Significant figures are an important concept in ensuring the precision and accuracy of numerical results in scientific calculations. They represent the digits in a number that carry meaningful information about its precision. For example, in the original solution's calculation of gravitational force, the final result was rounded to three significant figures:\[F \approx 1.11 \times 10^{-7} \mathrm{~N}\]When rounding using significant figures, you usually match the lowest number of significant figures from the data you have. Here, the weights were given as \(4.00\) kN, suggesting three significant figures. Therefore, presenting the gravitational force with three significant figures maintains consistency and reflects the accuracy level of the input values. Proper use of significant figures helps communicate the certainty of measurements and ensures consistency throughout calculations, aiding in clearer interpretation and validation of results.