Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity

Section Learning Objectives

By the end of this section, you will be able to do the following:

Explain Newton’s law of universal gravitation and compare it to Einstein’s theory of general relativity
Perform calculations using Newton’s law of universal gravitation

Section Key Terms

Einstein’s theory of general relativity

gravitational constant

Newton’s universal law of gravitation

Concepts Related to Newton’s Law of Universal Gravitation

Sir Isaac Newton was the first scientist to precisely define the gravitational force, and to show that it could explain both falling bodies and astronomical motions. See Figure 7.7. But Newton was not the first to suspect that the same force caused both our weight and the motion of planets. His forerunner, Galileo Galilei, had contended that falling bodies and planetary motions had the same cause. Some of Newton’s contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, had also made some progress toward understanding gravitation. But Newton was the first to propose an exact mathematical form and to use that form to show that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas. This theoretical prediction was a major triumph. It had been known for some time that moons, planets, and comets follow such paths, but no one had been able to propose an explanation of the mechanism that caused them to follow these paths and not others.

A painting shows a man sitting under a tree, from which an apple is falling.

Figure 7.7 The popular legend that Newton suddenly discovered the law of universal gravitation when an apple fell from a tree and hit him on the head has an element of truth in it. A more probable account is that he was walking through an orchard and wondered why all the apples fell in the same direction with the same acceleration. Great importance is attached to it because Newton’s universal law of gravitation and his laws of motion answered very old questions about nature and gave tremendous support to the notion of underlying simplicity and unity in nature. Scientists still expect underlying simplicity to emerge from their ongoing inquiries into nature.

The gravitational force is relatively simple. It is always attractive, and it depends only on the masses involved and the distance between them. Expressed in modern language, Newton’s universal law of gravitation states that every object in the universe attracts every other object with a force that is directed along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This attraction is illustrated by Figure 7.8.

A figure shows a small circle and a large circle. A black line goes from one circle to the other. A red line with an arrow extends outward from each circle. The red lines have arrows on one end, pointing away from the circles, towards the midpoint of the space in between the circles.

Figure 7.8 Gravitational attraction is along a line joining the centers of mass (CM) of the two bodies. The magnitude of the force on each body is the same, consistent with Newton’s third law (action-reaction).

For two bodies having masses m and M with a distance r between their centers of mass, the equation for Newton’s universal law of gravitation is

F = G \frac{m M}{r^{2}} F = G \frac{m M}{r^{2}}

where F is the magnitude of the gravitational force and G is a proportionality factor called the gravitational constant. G is a universal constant, meaning that it is thought to be the same everywhere in the universe. It has been measured experimentally to be $G = 6.673 \times 10^{- 11} N \cdot m^{2} {/kg}^{2} G = 6.673 \times 10^{- 11} N \cdot m^{2} {/kg}^{2}$ .

If a person has a mass of 60.0 kg, what would be the force of gravitational attraction on him at Earth’s surface? G is given above, Earth’s mass M is 5.97 × 10²⁴ kg, and the radius r of Earth is 6.38 × 10⁶ m. Putting these values into Newton’s universal law of gravitation gives

F = G \frac{m M}{r^{2}} = (6.673 \times 10^{- 11} \frac{N \cdot m^{2}}{{kg}^{2}}) (\frac{(60.0 kg) (5.97 \times 10^{24} kg)}{{(6.38 \times 10^{6} m)}^{2}}) = 584 N F = G \frac{m M}{r^{2}} = (6.673 \times 10^{- 11} \frac{N \cdot m^{2}}{{kg}^{2}}) (\frac{(60.0) (5.97 \times 10^{24} kg)}{{(6.38 \times 10^{6} m)}^{2}}) = 584

We can check this result with the relationship: $F = m g = (60 kg) (9.8 {m/s}^{2}) = 588 N F = m g = (60$

You may remember that g, the acceleration due to gravity, is another important constant related to gravity. By substituting g for a in the equation for Newton’s second law of motion we get $F = m g F = m g$ . Combining this with the equation for universal gravitation gives

m g = G \frac{m M}{r^{2}} m g = G \frac{m M}{r^{2}}

Cancelling the mass m on both sides of the equation and filling in the values for the gravitational constant and mass and radius of the Earth, gives the value of g, which may look familiar.

g = G \frac{M}{r^{2}} = (6.67 \times 10^{- 11} \frac{N \cdot m^{2}}{{kg}^{2}}) (\frac{5.98 \times 10^{24} kg}{{(6.38 \times 10^{6} m)}^{2}}) = 9.80 {m/s}^{2} g = G \frac{M}{r^{2}} = (6.67 \times 10^{- 11} \frac{N \cdot m^{2}}{{kg}^{2}}) (\frac{5.98 \times 10^{24} kg}{{(6.38 \times 10^{6} m)}^{2}}) = 9.80 {m/s}^{2}

This is a good point to recall the difference between mass and weight. Mass is the amount of matter in an object; weight is the force of attraction between the mass within two objects. Weight can change because g is different on every moon and planet. An object’s mass m does not change but its weight mg can.

Virtual Physics

Gravity and Orbits

Move the sun, Earth, moon and space station in this simulation to see how it affects their gravitational forces and orbital paths. Visualize the sizes and distances between different heavenly bodies. Turn off gravity to see what would happen without it!

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