Suppose astronomers discover a \(3-M_{\text {sun }}\) black hole located a few light-years from Earth. Should they be concerned that its tremendous gravitational pull will lead to Earth's untimely demise?

Newton’s Law of Gravitation is a crucial principle in understanding the forces that celestial objects exert on each other. It states that every point mass attracts every other point mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The mathematical representation is: \[ F = G \frac{m_1 m_2}{r^2} \]Here, \( F \) represents the gravitational force, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( r \) is the distance between them. This law helps us calculate the gravitational force exerted by a black hole on Earth. Using this understanding, we can compare this force with the gravitational influence we experience from the Sun.

Mass is a fundamental property of matter, indicating how much matter an object contains. In Newton’s law, mass is a critical variable because the gravitational force depends directly on the masses of the two objects involved. When dealing with celestial bodies like black holes and planets:

• The mass of the black hole is often given in terms of the Sun’s mass, for example, \( 3M_{\text{sun}} \), meaning three times the Sun's mass.
• The mass of Earth is a well-known constant, approximately \( 5.972 \times 10^{24} \text{ kg} \).

Understanding these masses allows us to calculate the gravitational force using Newton’s Law of Gravitation.

Distance is another crucial factor in Newton’s Law of Gravitation. The force of gravity decreases rapidly as the distance between two objects increases because the gravitational force is inversely proportional to the square of the distance \( r \) between them. This means that if the distance between two objects doubles, the gravitational force is reduced to a quarter. In our example:

• The black hole is four light-years away from Earth.
• Each light-year is approximately \( 9.46 \times 10^{15} \text{ meters} \).

Given these distances, we convert light-years to meters to plug into the gravitation formula.

A light-year is a unit of distance used in astronomy to express the vast distances between stars and other celestial bodies. It is defined as the distance that light travels in one year. For reference:

• One light-year is about \( 9.46 \times 10^{15} \text{ meters} \).
• So, four light-years is approximately \( 3.784 \times 10^{16} \text{ meters} \).

By converting light-years to meters, we can accurately use this distance in our gravitational force calculations.

The gravitational constant, denoted as \( G \), is a fundamental physical constant involved in the calculation of gravitational force between two bodies. Its value is approximately \( 6.67 \times 10^{-11} \text{ N(m}^2\text{/kg}^2 \). The gravitational constant makes sure that our units of mass and distance (in kilograms and meters, respectively) balance correctly to give us a force in Newtons. In our calculations:

• Using \( G \) ensures that the gravitational force between Earth and the black hole can be accurately determined.
• Without \( G \), we wouldn’t be able to relate the mass and distance to produce a meaningful force.