14.4
Gravity between Spherical Bodies
Newton's law of gravitation describes the gravitational force between any two point masses. However, for extended spherical objects like the Earth, th…
An apple released from a tree naturally tends to fall on the ground. This downward force was thought to be unique to the Earth.
However, Sir Isaac Newton reasoned that this force also causes the Moon to follow its path around the Earth.
Assuming that the Earth and the Moon have spherically symmetric mass distributions, their masses can be considered to be concentrated at their respective centers.
The Earth's mass is 5.97 × 1024 kilograms, and the Moon's mass is 7.35 × 1022 kilograms. The average distance between their centers is 3.84 × 108 meters.
Recall that the gravitational constant equals 6.67 × 10−11 newton meter squared per kilogram squared.
Substituting the known values in the gravitational force equation, the gravitational force between the Earth and the Moon is estimated to be 1.98 × 1020 newtons.
Similarly, considering the Sun as a spherically symmetric body, substituting its mass and its distance from the Earth, the gravitational force between the Sun and the Earth can be calculated.
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Q1: Why can Newton's law of gravitation be applied to spherical bodies like Earth and the Moon?
Newton's law of gravitation applies to spherical bodies because their masses can be treated as concentrated at their respective centers. This assumption holds for both hollow and solid spheres due to gravitational potential energy equivalence. The mathematical proof demonstrates that a spherical mass distribution produces the same gravitational effect as a point mass located at the sphere's center.
Q2: What is the gravitational force between Earth and the Moon?
Using Newton's law of gravitation with Earth's mass of 5.97 × 10²⁴ kilograms, the Moon's mass of 7.35 × 10²² kilograms, and their average center-to-center distance of 3.84 × 10⁸ meters, the gravitational force is approximately 1.98 × 10²⁰ newtons. This calculation assumes spherically symmetric mass distributions and uses the gravitational constant of 6.67 × 10⁻¹¹ newton meter squared per kilogram squared.
Q3: How does the shell theorem prove that spherical masses act as point masses?
The shell theorem is proven by analyzing gravitational potential energy between a hollow sphere and a point mass. By integrating the contributions of infinitesimal rings across the sphere's surface, the result equals the potential energy between two point masses separated by the same distance. Since force derives from potential energy, this proof extends to gravitational forces between any two spherical objects.
Q4: How did Newton connect falling apples to the Moon's orbit?
Newton reasoned that the downward force causing an apple to fall is the same gravitational force that keeps the Moon in orbit around Earth. He recognized that gravity is not unique to Earth but acts universally between all massive bodies. This insight unified terrestrial and celestial mechanics under a single gravitational principle applicable to spherically symmetric bodies.
Q5: Can you calculate the gravitational force between the Sun and Earth using the same method?
Yes, the same approach applies to the Sun and Earth. By treating the Sun as a spherically symmetric body and substituting its mass and distance from Earth into Newton's gravitational force equation, you can calculate their gravitational force. The method remains consistent because both the Sun and Earth have spherically symmetric mass distributions, allowing their masses to be concentrated at their centers.
Q6: What role does the distance between sphere centers play in calculating gravitational force?
The distance between sphere centers is critical in Newton's gravitational force equation. For spherical bodies, this distance is measured from center to center, not surface to surface. This center-to-center measurement is valid because the entire mass of each sphere can be treated as concentrated at its center, making the calculation straightforward and accurate for planetary and lunar systems.
Q7: Why is the assumption of concentrated mass valid for extended spherical objects?
The concentrated mass assumption is valid because gravitational potential energy calculations for hollow spheres yield identical results whether the mass is distributed across the sphere's surface or concentrated at its center. This mathematical equivalence, proven through integration of ring contributions, ensures that Newton's law of gravitation produces accurate gravitational forces for extended spherical solids like planets and moons.