Problem 4 - Classical Mechanics - An Undergraduate Text

Question

The internal gravitational potential energy of a system of masses is sometimes called the self energy of the system. (The reference configuration is taken to be one in which the particles are all a great distance from each other.) Show that the self energy of a uniform sphere of mass \(M\) and radius \(R\) is \(-3 M^{2} G / 5 R\). [Imagine that the sphere is built up by the addition of successive thin layers of matter brought in from infinity.]

Step-by-Step Solution

Answer

The self-energy of a uniform sphere of mass \(M\) and radius \(R\) is \(-\frac{3GM^2}{5R}\).

Step 1: Identifying Variables

Here, \(M\) is the total mass of the sphere, \(R\) is radius of the sphere, and \(G\) is gravitational constant. We have to find the self-energy \(U\). The mass of an infinitesimally thin layer at radius \(r\) with thickness \(dr\) can be written as \(\delta m = \rho 4 \pi r^2 dr\), where \(\rho\) is the density of the sphere and it equals to \(M / (4/3 \pi R^3)\). Since the volume of thin layer \(V = 4 \pi r^2 dr\), the mass can also be written as \(\delta m = M V / V_{sphere}\), where \(V_{sphere} = 4/3 \pi R^3\).

Step 2: Calculating the Work Done

Then we can calculate the work done \(dW\) to bring this mass \(\delta m\) from infinity to the point \(r\) in the gravitational field of mass \(m(r) = M r^3 / R^3\), which is inside the sphere of radius \(r\). The work done will be \(dW = - G m(r) \delta m / r = - G M^2 r^2 dr / R^6\).

Step 3: Total Energy

To find the total gravitational potential energy or the self-energy, integrate the \(dW\) from \(0\) to \(R\), this gets \[U = \int_{0}^{R}{- \frac{G M^2 r^2 dr}{R^6}} = -\frac{3GM^2}{5R}.\] The negative sign indicates the energy is being conserved.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy that an object possesses because of its position in a gravitational field. It is a form of potential energy stored by an object due to its height above the ground, or in the case of our uniform sphere, due to its mass being in proximity to other mass in space.
The concept arises from the force of gravity that acts on the object or particles within the system. This force does work on the object when it moves towards or away from the source of the gravitational field, changing its potential energy in the process.

• For our uniform sphere, this "self energy" refers to the energy needed to assemble the sphere by moving thin layers of mass from infinity to form the sphere.
• It's important to note that gravitational potential energy is considered negative in physics. This is because work must be done against the gravitational force to remove an object from a massive body like Earth, or our sphere.

By calculating this energy, we can understand how much energy is bound within a gravitational system.

Uniform Sphere

A uniform sphere is a sphere that has a consistent density throughout its entire volume. This means that any slice taken from the sphere, regardless of location, has the same mass per unit volume. The concept of a uniform sphere is very helpful when calculating gravitational forces and potential energy, as it simplifies the mathematical model of the sphere.

• In physics, assuming uniformity allows for more straightforward calculations, because every small part of the sphere contributes equally to the overall mass.
• This is key when calculating the gravitational potential energy of a sphere as you're essentially "building" the sphere layer by layer.

Each layer contributes a certain amount of work that must be done to bring it from infinity and add it to the growing sphere without changing the density distribution.

Mass Density

Mass density, often simply referred to as density, is the amount of mass per unit volume of a substance. For a uniform sphere, this is a crucial term, as it determines how mass is distributed throughout the sphere which in turn affects its gravitational properties.

• The density \( \rho \) of the sphere is calculated by dividing the total mass \( M \) by the volume of the sphere \( \frac{4}{3} \pi R^3 \).
• Density remains constant throughout the sphere in a uniform sphere model, simplifying calculations as each layer added maintains a consistent density.

Understanding mass density is essential since it directly influences the work needed to bring additional layers from a point at infinity to their respective positions in the sphere's composition.

Work-Energy Principle

The work-energy principle is a fundamental concept in physics that relates the amount of work done on an object to its change in energy. In our case, it helps us understand how the work done in assembling layers of the sphere relates to the gravitational potential energy of the sphere as a whole.

• When considering the assembly of a sphere, we calculate the amount of work (\( dW \)) needed to add each infinitesimally thin layer of mass \( \delta m \) to the sphere.
• The work done to assemble these layers from infinity is negative, signifying the release of energy as the layers move into the gravitational field of the sphere.

The work-energy principle shows us that the total gravitational potential energy, or self-energy, is the sum of all the small amounts of work done during the assembly.