(b) Using Gauss's law, determine the gravitational field g(r) for all points in the regions 0R
(b) Using Gauss's law, determine the gravitational field g(r) for all points in the regions 0R
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![**Solid Sphere Volume Mass Density**
Consider a solid sphere (e.g., a planet) with mass \( M \) and radius \( R \). The volume mass density for this planet is given by:
\[
\rho(r) =
\begin{cases}
A \left(1 - \frac{r^2}{R^2}\right) & \text{for } r \leq R \\
0 & \text{for } r > R
\end{cases}
\]
where \( A \) is a constant with the units of kg/m\(^3\).
**Explanation:**
This expression defines the density \(\rho(r)\) as a function of the radial distance \( r \) from the center of the sphere.
- For \( r \leq R \), the density decreases with \( r^2 \), indicating that the density is highest at the center and decreases towards the surface.
- For \( r > R \), the density is zero, which reflects the fact that there is no material beyond the sphere's surface.
This is an example of a model used to understand how mass might be distributed within a solid celestial object, like a planet.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56e8b022-9dd8-4fb4-b3ca-fd6588a1a301%2Fc96d8328-eaa4-4754-9a94-0cdd304661bc%2Fk77nxk_processed.png&w=3840&q=75)
Transcribed Image Text:**Solid Sphere Volume Mass Density**
Consider a solid sphere (e.g., a planet) with mass \( M \) and radius \( R \). The volume mass density for this planet is given by:
\[
\rho(r) =
\begin{cases}
A \left(1 - \frac{r^2}{R^2}\right) & \text{for } r \leq R \\
0 & \text{for } r > R
\end{cases}
\]
where \( A \) is a constant with the units of kg/m\(^3\).
**Explanation:**
This expression defines the density \(\rho(r)\) as a function of the radial distance \( r \) from the center of the sphere.
- For \( r \leq R \), the density decreases with \( r^2 \), indicating that the density is highest at the center and decreases towards the surface.
- For \( r > R \), the density is zero, which reflects the fact that there is no material beyond the sphere's surface.
This is an example of a model used to understand how mass might be distributed within a solid celestial object, like a planet.
Transcribed Image Text:**(b)** Using Gauss's law, determine the gravitational field \( \vec{g}(r) \) for all points in the regions \( 0 < r < R \) and \( r > R \).
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