A uniform, spherical, $1000.0\text{ kg}$ shell has a radius of $5.00\text{ m}$. Sketch a qualitative graph of the magnitude of the gravitational force this sphere exerts on a point mass m as a function of the distance $r$ of $m$ from the center of the sphere. Include the region from $r=0$ to $r\to \infty$.

Verified step by step guidance

1. Understand the concept of gravitational force exerted by a spherical shell: According to the shell theorem, a uniform spherical shell of mass exerts no gravitational force on a point mass located inside the shell. For a point mass outside the shell, the shell behaves as if all its mass were concentrated at its center.
2. Identify the regions to consider: The problem requires analyzing the gravitational force from r = 0 to r -> ∞. This means considering two main regions: inside the shell (0 ≤ r < 5.00 m) and outside the shell (r ≥ 5.00 m).
3. For the region inside the shell (0 ≤ r < 5.00 m): According to the shell theorem, the gravitational force exerted by the shell on a point mass inside is zero. Therefore, the graph should show zero force for this region.
4. For the region outside the shell (r ≥ 5.00 m): Use Newton's law of universal gravitation to determine the force. The gravitational force F can be expressed as: $F=\frac{GmM}{r^2}$, where G is the gravitational constant, m is the point mass, M is the mass of the shell, and r is the distance from the center of the shell.
5. Sketch the graph: For r < 5.00 m, the graph is a horizontal line at zero. For r ≥ 5.00 m, the graph shows a curve that decreases with increasing r, following the inverse square law. The graph should start at a maximum value at r = 5.00 m and approach zero as r approaches infinity.

Key Concepts

Gravitational Force Inside a Spherical Shell

According to the shell theorem, a uniform spherical shell of mass exerts no net gravitational force on a point mass located inside it. This means that for any point mass m located at a distance r less than the radius of the shell, the gravitational force is zero. This is crucial for understanding the behavior of the force as the point mass moves from the center to the surface of the sphere.

Gravitational Force Outside a Spherical Shell

For a point mass located outside a spherical shell, the shell can be treated as if all its mass were concentrated at its center. The gravitational force exerted by the shell on the point mass is given by Newton's law of universal gravitation: F = G(Mm)/r², where M is the mass of the shell, m is the mass of the point, r is the distance from the center of the shell, and G is the gravitational constant. This concept helps in sketching the force for r greater than the shell's radius.

Qualitative Graphing of Gravitational Force

To sketch the gravitational force as a function of distance, consider the behavior inside and outside the shell. Inside (r < 5 m), the force is zero. At the surface (r = 5 m), the force begins to increase, following the inverse square law as r increases beyond the shell's radius. The graph should show a flat line at zero for r < 5 m, then a curve decreasing with 1/r² for r > 5 m, approaching zero as r approaches infinity.