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Chapter 6: Problem 756

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

Expert verified
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

01

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.
02

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.
03

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.

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