Understanding Newton's Law of Gravitation

Newton's Law of Gravitation explains how gravity works between two objects with mass.

Here’s the main idea:

The gravitational force ($F_g$) between two objects, which we’ll call (m_1) and (m_2), can be calculated using this formula:

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

In this formula:

• (G) is a constant, and it’s about (6.674 \times 10^{-11} N m^2/kg^2).
• (r) is the distance between the centers of the two objects.

When we think about objects moving in a circle, like a satellite orbiting Earth, we can also talk about weight.

The weight (W) of an object in circular motion is the gravitational force acting on it. For example, if a satellite has mass (m), the force that keeps it moving in a circle comes from gravity:

$W = m \cdot a_c = m \cdot \frac{v^2}{r}$

In this equation:

• (a_c) is the centripetal acceleration, which is the force that keeps the object moving in a circle.
• (v) is the speed of the object.
• (r) is the distance from the center of the circle.

When we set the gravitational force equal to this centripetal force, we get this equation:

$G \frac{M m}{r^2} = m \frac{v^2}{r}$

From this, we can simplify to find how speed relates to gravity:

$v^2 = \frac{G M}{r}$

This shows us that gravity plays a big role in the weight of objects that move in a circle.

Overall, it’s gravity that helps keep satellites and other objects in orbit!