Question

Which one of the following graphs represents correctly the variation of the gravitational field with the distance (r) from the center of spherical shell of mass \(\mathrm{M}\) and radius a

Short answer

A correct graph representing the variation of the gravitational field with distance (r) from a center of a spherical shell of mass M and radius a should have the following attributes: 1. The gravitational field is 0 for \(r < a\) (inside the shell) because a uniform spherical shell generates no net gravitational force on particles inside it. 2. The gravitational field decreases as the inverse-square of the distance \(r\) for \(r > a\) (outside the shell) using the formula: \(g(r) = G\frac{M}{r^2}\), where G is the gravitational constant. Comparing the given graphs, choose the graph that exhibits these two attributes.

Step-by-step solution

Gravitational field within a spherical shell

To find the gravitational field within the spherical shell, we can use the fact that a uniform spherical shell creates no net gravitational force on particles inside it. That means the gravitational field inside the shell is equal to 0 for r < a, where r is the distance from the center of the spherical shell, and a is the radius of the shell.

Gravitational field outside the spherical shell

For points outside the spherical shell (r > a), we can treat the shell as if its entire mass M is concentrated at its center. In this region, the gravitational field follows the same inverse-square law as the point mass (M): \[ g(r) = G\frac{M}{r^2}, \] where G is the gravitational constant, and r is the distance from the center of the shell.

Comparison with graphs

Now, compare the behavior of the gravitational field we found in Step 1 and Step 2 with the given graphs. The correct graph should show the following attributes: 1. The gravitational field should be 0 for r < a (inside the shell). 2. The gravitational field should decrease as the inverse-square of the distance r for r > a (outside the shell). Check each graph to see which one meets the above criteria. The graph that matches the behavior of the gravitational field in the shell and outside the shell will be the correct representation.

Key concepts

Spherical Shell

When we talk about a spherical shell, we imagine a three-dimensional object resembling a thick hollow bubble. Think of it like the surface of a baseball, without anything inside but air. This type of shell is symmetrical in all directions from its center, meaning every point on its surface is equidistant from the central point.
This symmetry plays a critical role in how gravitational forces behave around the shell.

• The shell is defined by its mass \(M\) and radius \(a\).
• For analyzing gravitational effects, it can often be treated as a point mass at its center when considering forces beyond the surface.

Gravitational fields around such structures depend heavily on the distance from the center of the shell, which is beautifully explained by the inverse-square law.

Inverse-square Law

The inverse-square law is a principle in physics that highlights how certain fundamental forces, such as gravity, diminish with distance. Specifically for gravity, this law states that the force acting between two objects is inversely proportional to the square of the distance separating them.

Thus, if you double the distance between two masses, the gravitational force becomes one-fourth as strong. Mathematically, this can be expressed with the formula:\[ g(r) = G\frac{M}{r^2} \] where:

• \(g(r)\) is the gravitational field strength at a distance \(r\).
• \(G\) is the gravitational constant.
• \(M\) is the mass creating the gravitational field.

This law is crucial when calculating how gravity behaves outside a spherical shell, as it directly impacts how the gravitational field diminishes with increasing distance from the shell's center.

Gravitational Field Inside Shell

The gravitational field inside a spherical shell is an interesting phenomenon because, surprisingly, it is zero. This occurs due to the shell's symmetry.

• Every particle within the shell is equally attracted to every part of the shell.
• The gravitational forces in every direction cancel each other out perfectly.

Consequently, an object placed anywhere inside the shell, no matter how far from the shell's center, experiences no net gravitational pull. This is why, for distances \(r < a\), the gravitational field is zero.

Understanding this concept helps in simplifying complex gravitational problems, especially those involving objects within large hollow structures.

Gravitational Field Outside Shell

For distances greater than the radius of the spherical shell \( (r > a) \), the shell can be treated as if all its mass is concentrated at a central point. This allows the application of the inverse-square law to find the gravitational field outside the shell.
Here, the gravitational field strength decreases with the square of the distance from the center. Mathematically, the field is expressed as: \[ g(r) = G\frac{M}{r^2} \] The assumptions behind this calculation rely heavily on the symmetry of the shell and the uniform distribution of its mass. This principle simplifies real-world calculations where large celestial bodies like planets and stars can be approximated as spherical shells for understanding their gravitational effects on surrounding objects.
For \(r > a\), a graph of \(g(r)\) would demonstrate a curve that drops off sharply as you move further from the shell, consistently following the inverse-square law.