Newton's Law of Universal Gravitation

Newton’s Law of Universal Gravitation

Newton’s law of universal gravitation connects motion on Earth with the motion of planets and other celestial bodies. Newton proposed that the force keeping planets in orbit around the Sun is the same gravitational force that makes objects fall toward Earth. In this view, orbiting objects are continually falling toward the body they orbit, but their sideways speed keeps them from hitting it.

Orbits as Continuous Free Fall

Newton illustrated orbital motion with a thought experiment involving a cannon on a high mountain:

• A cannonball fired at low speed travels a short distance before falling to the ground.
• Greater launch speed makes it travel farther before it lands.
• At a sufficiently high speed, the ball would fall toward Earth at the same rate that Earth’s surface curves away beneath it.
• It would therefore keep falling without reaching the ground, entering orbit.

Satellites and space stations follow this principle: they move fast enough to remain in continuous free fall around Earth. The same general idea applies to planets orbiting the Sun.

The Universal Gravitational Force

Every object with mass exerts a gravitational force on every other object with mass. The force between two objects is described by:

[ F = G\frac{m_1m_2}{r^2} ]

where:

• (F) is the gravitational force.
• (m_1) and (m_2) are the objects’ masses.
• (r) is the distance between their centers.
• (G) is the universal gravitational constant, approximately (6.67 \times 10^{-11}\ \mathrm{N\,m^2/kg^2}).

The force increases with either object’s mass and decreases with the square of the distance between them. Henry Cavendish determined the value of (G) experimentally about a century after Newton’s work.

For calculating gravitational attraction, objects can be treated as point masses, with the relevant distance measured between their centers.

Mutual Forces and Motion

• Two bodies pull on each other with forces that are equal in magnitude.
• Equal forces do not necessarily cause equal accelerations because (F = ma): the less massive object accelerates more.
• In the Earth–Moon system, both bodies orbit their shared center of mass. Because Earth is much more massive, that center lies within Earth, making it appear that the Moon simply orbits Earth.
• Earth and a falling apple likewise pull on each other, but Earth’s much greater mass makes its resulting acceleration imperceptibly small.

Why Objects Fall at the Same Rate

Ignoring air resistance, objects of different masses fall toward Earth with the same acceleration. A more massive object experiences a stronger gravitational force, but it also has proportionally greater inertia.

Using Newton’s second law and the gravitational-force equation:

[ ma = G\frac{mM_{\text{Earth}}}{r^2} ]

The falling object’s mass cancels, giving:

[ a = G\frac{M_{\text{Earth}}}{r^2} ]

Thus, the acceleration does not depend on the falling object’s mass. Near Earth’s surface, it is approximately (9.8\ \mathrm{m/s^2}).

Limits and Later Developments

Newton’s theory successfully described a wide range of terrestrial and celestial motion, but it did not explain how gravity acts across empty space. Later, gravity was described as a field associated with matter. Einstein’s general theory of relativity provided a more sophisticated account, connecting gravity with the structure of space and helping scientists understand the formation and behavior of planets, stars, and galaxies. Gravity remains an active area of study.

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• Professor Dave — narrator and instructor.