Gravity: The Universal Pull

The force that drops an apple also steers the Moon. It acts between you and the person sitting next to you, too, just far too weakly to notice.

The law of universal gravitation#

Any two bodies attract each other simply because they have mass. For two point masses $m_1$ and $m_2$ a distance $r$ apart, the attraction has magnitude

$G$ is a universal constant, the same for apples, planets and galaxies. Two features matter most: the force grows with each mass, and it falls off as the inverse square of distance. Double the separation and the pull drops to a quarter.

Big bodies: treat spheres as points#

Real objects are not points. To find the pull of one extended body on another, you would have to add up the force of every particle of the first body on every particle of the second, as vectors. Even two bodies of three particles each produce nine forces to add; real bodies contain trillions of trillions.

Treating the bodies as continuous turns that sum into an integral, and for one important case the answer is beautifully simple (we prove it in a later chapter):

Gravity at the Earth’s surface#

The Earth’s pull on objects is called gravity. Model the Earth as a uniform sphere of mass $M \approx 6\times10^{24}$ kg and radius $R \approx 6400$ km. A small body of mass $m$ near the surface feels a force $GMm/R^2$ toward the Earth’s centre, the direction we call “vertically downward.” Group the constants:

The quantity $g$ has the dimensions of acceleration, and in the next chapter we will see that it is exactly the acceleration of any body falling freely near the surface. It is called the acceleration due to gravity. We often round it to 10 m/s² to keep arithmetic simple. So the Earth pulls on a small body of mass $m$ with a force $mg$, straight down. That force is the body’s weight.