The Law of Universal Gravitation

Objects with mass feel an attractive force that is proportional to their masses and inversely proportional to the square of the distance.

While an apple might not have struck Sir Isaac Newton’s head as myth suggests, the falling of one did inspire Newton to one of the great discoveries in mechanics: The Law of Universal Gravitation. Pondering why the apple never drops sideways or upwards or any other direction except perpendicular to the ground, Newton realized that the Earth itself must be responsible for the apple’s downward motion.

Theorizing that this force must be proportional to the masses of the two objects involved, and using previous intuition about the inverse-square relationship of the force between the earth and the moon, Newton was able to formulate a general physical law by induction.

The Law of Universal Gravitation states that every point mass attracts every other point mass in the universe by a force pointing in a straight line between the centers-of-mass of both points, and this force is proportional to the masses of the objects and inversely proportional to their separation This attractive force always points inward, from one point to the other. The Law applies to all objects with masses, big or small. Two big objects can be considered as point-like masses, if the distance between them is very large compared to their sizes or if they are spherically symmetric. For these cases the mass of each object can be represented as a point mass located at its center-of-mass.

While Newton was able to articulate his Law of Universal Gravitation and verify it experimentally, he could only calculate the relative gravitational force in comparison to another force. It wasn’t until Henry Cavendish’s verification of the gravitational constant that the Law of Universal Gravitation received its final algebraic form:

[latex]\displaystyle \text{F} = \text{G}\frac{\text{Mm}}{\text{r}^2}[/latex]

where [latex]\text{F}[/latex] represents the force in Newtons, [latex]\text{M}[/latex] and [latex]\text{m}[/latex] represent the two masses in kilograms, and [latex]\text{r}[/latex] represents the separation in meters. [latex]\text{G}[/latex] represents the gravitational constant, which has a value of [latex]6.674\cdot 10^{-11} \text{N}\text{(m/kg)}^2[/latex]. Because of the magnitude of [latex]\text{G}[/latex], gravitational force is very small unless large masses are involved.

Weight of the Earth

When the bodies have spatial extent, gravitational force is calculated by summing the contributions of point masses which constitute them.

Newton’s law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses, and inversely proportional to the square of the distance between them.

In modern language, the law states the following: Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them:

[latex]\displaystyle \text{F} = \text{G}\frac{\text{m}_{1}\text{m}_{2}}{\text{r}^{2}}[/latex]

where [latex]\text{F}[/latex] is the force between the masses, [latex]\text{G}[/latex] is the gravitational constant, [latex]\text{m}_1[/latex] is the first mass, [latex]\text{m}_2[/latex] is the second mass and [latex]\text{r}[/latex] is the distance between the centers of the masses.

If the bodies in question have spatial extent (rather than being theoretical point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses which constitute the bodies. In the limit, as the component point masses become “infinitely small”, this entails integrating the force (in vector form, see below) over the extents of the two bodies.

In this way it can be shown that an object with a spherically-symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object’s mass were concentrated at a point at its center.

For points inside a spherically-symmetric distribution of matter, Newton’s Shell theorem can be used to find the gravitational force. The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance [latex]\text{r}_0[/latex] from the center of the mass distribution:

1. The portion of the mass that is located at radii [latex]\text{r}<\text{r}_0[/latex] causes the same force at [latex]\text{r}_0[/latex] as if all of the mass enclosed within a sphere of radius [latex]\text{r}_0[/latex] was concentrated at the center of the mass distribution (as noted above).
2. The portion of the mass that is located at radii [latex]\text{r}>\text{r}_0[/latex] exerts no net gravitational force at the distance [latex]\text{r}_0[/latex] from the center. That is, the individual gravitational forces exerted by the elements of the sphere out there, on the point at [latex]\text{r}_0[/latex], cancel each other out.

As a consequence, for example, within a shell of uniform thickness and density there is no net gravitational acceleration anywhere within the hollow sphere. Furthermore, inside a uniform sphere the gravity increases linearly with the distance from the center; the increase due to the additional mass is 1.5 times the decrease due to the larger distance from the center. Thus, if a spherically symmetric body has a uniform core and a uniform mantle with a density that is less than [latex]\frac{2}{3}[/latex] of that of the core, then the gravity initially decreases outwardly beyond the boundary, and if the sphere is large enough, further outward the gravity increases again, and eventually it exceeds the gravity at the core/mantle boundary.

The gravity of the Earth may be highest at the core/mantle boundary, as shown in Figure 1: