PHYS101: Introduction to Mechanics

What do aching feet, a falling apple, and the orbit of the Moon have in common? Each is caused by the gravitational force. Our feet are strained by supporting our weight – the force of Earth's gravity on us. An apple falls from a tree because of the same force acting a few meters above Earth's surface. And the Moon orbits Earth because gravity is able to supply the necessary centripetal force at a distance of hundreds of millions of meters. In fact, the same force causes planets to orbit the Sun, stars to orbit the center of the galaxy, and galaxies to cluster together.

Gravity is another example of underlying simplicity in nature. It is the weakest of the four basic forces found in nature, and in some ways the least understood. It is a force that acts at a distance, without physical contact, and is expressed by a formula that is valid everywhere in the universe, for masses and distances that vary from the tiny to the immense.

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 6.20. 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 a mechanism that caused them to follow these paths and not others.

Figure 6.20 According to early accounts, Newton was inspired to make the connection between falling bodies and astronomical motions when he saw an apple fall from a tree and realized that if the gravitational force could extend above the ground to a tree, it might also reach the Sun. The inspiration of Newton's apple is a part of worldwide folklore and may even be based in fact. 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. Stated in modern language, Newton's universal law of gravitation states that every particle in the universe attracts every other particle with a force 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.

Figure 6.21 Gravitational attraction is along a line joining the centers of mass of these two bodies. The magnitude of the force is the same on each, consistent with Newton's third law.

The bodies we are dealing with tend to be large. To simplify the situation we assume that the body acts as if its entire mass is concentrated at one specific point called the center of mass (CM), which will be further explored in Linear Momentum and Collisions. 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

$F=G \frac{mM}{r^2}$,

where $F$ is the magnitude of the gravitational force and $G$ is a proportionality factor called the gravitational constant. $G$ is a universal gravitational constant – that is, it is thought to be the same everywhere in the universe. It has been measured experimentally to be

$G=6.674×10^{−11} \frac{N⋅m^2}{kg^2}$

in SI units. Note that the units of $G$ are such that a force in newtons is obtained from $F=G \frac{mM}{r^2}$, when considering masses in kilograms and distance in meters. For example, two 1.000 kg masses separated by 1.000 m will experience a gravitational attraction of $6.674×10^{−11}\ N$. This is an extraordinarily small force. The small magnitude of the gravitational force is consistent with everyday experience. We are unaware that even large objects like mountains exert gravitational forces on us. In fact, our body weight is the force of attraction of the entire Earth on us with a mass of $6×10^{24}kg$.

Recall that the acceleration due to gravity g is about $9.80\ m/s^2$ on Earth. We can now determine why this is so. The weight of an object mg is the gravitational force between it and Earth. Substituting mg for $F$ in Newton's universal law of gravitation gives

$mg=G \frac{mM}{r^2}$,

where $m$ is the mass of the object, $M$ is the mass of Earth, and $r$ is the distance to the center of Earth (the distance between the centers of mass of the object and Earth). See Figure 6.22. The mass $m$ of the object cancels, leaving an equation for $g$:

$g=G \frac{M}{r^2}$.

Substituting known values for Earth's mass and radius (to three significant figures),

$g=\left(6.67 \times 10^{-11} \frac{\mathrm{N} \cdot \mathrm{m}^{2}}{\mathrm{~kg}^{2}}\right) \times \frac{5.98 \times 10^{24} \mathrm{~kg}}{\left(6.38 \times 10^{6} \mathrm{~m}\right)^{2}}$

and we obtain a value for the acceleration of a falling body:

$g=9.80\ m/s^2$.

Figure 6.22 The distance between the centers of mass of Earth and an object on its surface is very nearly the same as the radius of Earth, because Earth is so much larger than the object.

This is the expected value and is independent of the body's mass. Newton's law of gravitation takes Galileo's observation that all masses fall with the same acceleration a step further, explaining the observation in terms of a force that causes objects to fall – in fact, in terms of a universally existing force of attraction between masses.

In the following example, we make a comparison similar to one made by Newton himself. He noted that if the gravitational force caused the Moon to orbit Earth, then the acceleration due to gravity should equal the centripetal acceleration of the Moon in its orbit. Newton found that the two accelerations agreed "pretty nearly".

Source: Rice University, https://openstax.org/books/college-physics/pages/6-5-newtons-universal-law-of-gravitation
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