Problem 60 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A sphere of radius \(r_{1}\) has a concentric spherical cavity of radius \(r_{2}\) (Fig. 40\()\) . Assume this spherical shell of thickness \(r_{1}-r_{2}\) is uniform and has a total mass \(M .\) Show that the gravitational potential energy of a mass \(m\) at a distance \(r\) from the center of the shell \(\left(r>r_{1}\right)\) is given by $$U=-\frac{G m M}{r}$$

Step-by-Step Solution

Answer

The gravitational potential energy for mass \(m\) at distance \(r > r_1\) is \(U = -\frac{G m M}{r}\).

Step 1: Conceptual Understanding

When a mass \(m\) is outside a spherical shell, the gravitational force it feels is as if all the shell's mass were concentrated at the center. Thus, the entire mass \(M\) of the shell can be treated as a point mass at the center for a mass \(m\) located at a distance \(r > r_1\) from the center.

Step 2: Use the Gravitational Potential Energy Formula

The gravitational potential energy \(U\) for a point mass \(m\) due to another point mass \(M\) is given by the formula:\[ U = -\frac{G M m}{r} \]where \(G\) is the gravitational constant and \(r\) is the distance from the center of the shell to the mass \(m\).

Step 3: Apply the Formula

Since \(m\) is outside the spherical shell and feels the gravitational force as if \(M\) is concentrated at the center, we can directly apply the formula:\[ U = -\frac{G M m}{r} \]This formula accounts for the condition that the mass \(m\) is at a distance \(r > r_1\) from the center of the shell.

Key Concepts

Spherical Shell

A spherical shell is a geometric shape that resembles the outer layer of a hollow ball. It is defined between two radii, where the inner radius creates a cavity and the outer radius forms the complete shell. For example, consider a sphere of radius \(r_1\) with a concentric cavity having a radius \(r_2\). The space between these two radii is the spherical shell.
The intriguing property of a spherical shell in gravitational physics is linked to how it exerts gravitational force. **Gauss's Law** for gravity tells us something fascinating: a point outside a spherical shell, regarding the gravitational force, feels as if the entire shell's mass is concentrated at the center. This applies if the point is outside the shell, but inside the shell, the gravitational force is zero.

• This makes calculations simpler by treating the mass as if it were all compressed to its center.
• This also explains why objects inside the shell feel no net gravitational force from the shell itself.

Recognizing these properties can vastly simplify the equations we use to determine gravitational effects when dealing with spherical shells.

Gravitational Force

The gravitational force is the attractive force that exists between any two masses. Sir Isaac Newton introduced this concept with his universal law of gravitation. In basic terms, any two objects with mass will attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Mathematically, it is described by the equation:\[F = G \frac{m1 \cdot m2}{r^2}\]where:

• \(F\) is the gravitational force between the two masses,
• \(G\) is the gravitational constant \(6.674 \times 10^{-11} N \cdot (m/kg)^2\),
• \(m1\) and \(m2\) are the masses of the objects,
• and \(r\) is the distance between the centers of the two masses.

Using this equation, we can calculate the force experienced by any object due to another mass. In the context of a spherical shell, a mass outside the shell will experience the gravitational force as if the shell's entire mass were concentrated at a single point at its center. This simplifies the problem significantly when applying Newton's laws to gravitational problems.

Point Mass

In physics, the concept of a point mass refers to a hypothetical object that has mass but no dimensions—an idealization used to simplify problems. When a mass has a symmetrical shape, like a sphere, it can often be treated as a point mass for calculations involving gravitational forces or potential energy.
This simplification is particularly useful in situations where measuring distances from a point is more straightforward than accounting for the distribution of mass across an object. For example, when calculating the gravitational potential energy or forces exerted by and on spherical shells, we effectively use a point mass approach.

• The gravitational potential energy due to a larger mass on another can be estimated using the formula: \( U = -\frac{GMm}{r} \) where all mass is deemed to be concentrated at the center of the sphere.
• This approximation is valid if the one mass is entirely outside the region of the other, allowing us to simplify otherwise complex gravitational interactions.

By treating bodies as point masses, we streamline and neatly sidestep the intricacies of real-world mass distribution, making computations much more manageable in theoretical physics.