All of electrostatics follows from the 1 / r^2 character of...

All of electrostatics follows from the $1 / r^{2}$ character of Coulomb's law, together with the principle of superposition. An analogous theory can therefore be constructed for Newton's law of universal gravitation. What is the gravitational energy of a sphere, of mass $M$ and radius $R,$ assuming the density is uniform? Use your result to estimate the gravitational energy of the sun (look up the relevant numbers). The sun radiates at a rate of $3.86 \times 10^{26} \mathrm{W}$; if all this came from stored gravitational energy, how long would the sun last? [The sun is in fact much older than that, so evidently this is not the source of its power.]

Key Concepts

Inverse Square Law

The inverse square law states that certain forces, like gravity and electrostatic force, diminish in strength proportional to the square of the distance from the source. This foundational principle explains why both Coulomb’s law and Newton’s law of gravitation feature a 1/r² dependence, indicating that the field created by a point source spreads out uniformly over the surface of a sphere whose area increases with r².

Principle of Superposition

The principle of superposition asserts that in linear systems, the total effect produced by multiple sources is the sum of the effects that each source would produce individually. In the context of gravitational and electrostatic fields, this means that the net potential at a point due to a distribution of mass or charge is the algebraic sum of the potentials due to each individual element of the distribution.

Gravitational Binding Energy of a Uniform Sphere

Gravitational binding energy is the energy required to disperse an object’s mass to infinity against its own gravitational pull. For a uniform sphere, this energy is derived by integrating the contributions of all mass elements under mutual gravitational attraction, resulting in the formula U = -3GM²/(5R). This concept is essential for understanding the stability of self-gravitating bodies and the energy available from processes like gravitational collapse.

Energy-Power Relationship

The energy–power relationship connects the amount of stored energy with the rate at which it is expended. Power is defined as energy per unit time. By dividing the total gravitational binding energy of an object, such as a star, by its luminosity (power output), one can estimate how long that energy supply would last if it were the sole source powering the object. This concept is important in astrophysics for assessing energy budgets and lifetimes of celestial bodies.

Transcript

00:01 Hi there.

00:01 So for this problem, we are told that all of the electrostatics follows from the 1 over r square character of colum's law, together with the principle of superposition.

00:14 An analogous theory can therefore be constructed for newton's law of universal gravitation.

00:21 So the question in this is what is the gravitational energy of a sphere of mass capital m and radius capital? are assuming that the density is uniform.

00:34 We need to use your result to estimate the gravitational energy of the sum, and the sun radiates at a rate that is given 3 .86 times 10 to the 26 watts.

00:48 If all of this came from the stored gravitational energy, how long will the sun last? so, with that said, what we need to do to solve this problem is to compare newton's law of universal gravitation to colom's law.

01:05 So we know that newton's law of universal gravitation is the following.

01:10 The force is equal to minus, because we know that that is an attractive force, minus the gravitational constant g times the mass m1 times the mass m2, and this divided by the separation distance r to the square and this in the radial direction.

01:29 Now, the colom's law states that the force between two charges is equal to.

01:37 The product between the charges divided by four times pi times epsilon sub -zero times the radius, the separation, s -square in the radial direction.

01:49 Now, in this case, evidently, 1 over 4 times pi times epsilon sub -0 is, it makes the part of the acceleration due to gravity and the charge makes the part of the mass.

02:09 So the gravitational energy of a sphere is therefore the gravitational energy of a sphere is equal to 3 over 5 times the gravitational constant times the mass the mass squared divided by the radius r square...

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