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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.

Forces on two masses: All masses are attracted to each other. The force is proportional to the masses and inversely proportional to the square of the distance. Gravitational Attraction of Spherical Bodies: A Uniform Sphere The Shell Theorem states that a spherically symmetric object affects other objects as if all of its mass were concentrated at its center. The gravitational force on an object within a hollow spherical shell is zero.

Key Terms center of mass: The center of mass COM is the unique point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. Finding the gravitational force between three-dimensional objects requires treating them as points in space.

For highly symmetric shapes such as spheres or spherical shells, finding this point is simple. The Shell Theorem Isaac Newton proved the Shell Theorem, which states that: A spherically symmetric object affects other objects gravitationally as if all of its mass were concentrated at its center, If the object is a spherically symmetric shell i.

Given that a sphere can be thought of as a collection of infinitesimally thin, concentric, spherical shells like the layers of an onion , then it can be shown that a corollary of the Shell Theorem is that the force exerted in an object inside of a solid sphere is only dependent on the mass of the sphere inside of the radius at which the object is.

That is because shells at a greater radius than the one at which the object is, do not contribute a force to an object inside of them Statement 2 of theorem. Case 1: A hollow spherical shell The gravitational force acting by a spherically symmetric shell upon a point mass inside it, is the vector sum of gravitational forces acted by each part of the shell, and this vector sum is equal to zero.

Diagram used in the proof of the Shell Theorem: This diagram outlines the geometry considered when proving The Shell Theorem.

The surface area of a thin slice of the sphere is shown in color. Note: The proof of the theorem is not presented here. Interested readers can explore further using the sources listed at the bottom of this article.

We can use the results and corollaries of the Shell Theorem to analyze this case. Weight of the Earth When the bodies have spatial extent, gravitational force is calculated by summing the contributions of point masses which constitute 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.

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. 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.

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