Gravitation Force Due to Hollow and Solid Spheres

Gravitational force is a fundamental force of nature that attracts two bodies with mass. The gravitational force exerted by spherical bodies, such as planets and stars, is a common topic in physics. We will explore the gravitational forces due to hollow and solid spheres, which are idealized models often used in physics problems.

Gravitational Force Basics

Before diving into the specifics of hollow and solid spheres, let's review the basic formula for gravitational force. According to Newton's law of gravitation, the force ($F$) between two masses ($m_1$ and $m_2$) is given by:

$$ F = G \frac{m_1 m_2}{r^2} $$

where $G$ is the gravitational constant and $r$ is the distance between the centers of the two masses.

Gravitation Force Due to Solid Spheres

A solid sphere is a three-dimensional object with mass distributed uniformly throughout its volume. The gravitational force outside a solid sphere is as if all its mass were concentrated at its center. Inside a solid sphere, the gravitational force varies linearly with distance from the center, reaching zero at the center.

Outside a Solid Sphere

When an object is outside a solid sphere, the gravitational force it experiences is:

$$ F = G \frac{Mm}{r^2} $$

where $M$ is the mass of the sphere, $m$ is the mass of the object, and $r$ is the distance from the center of the sphere to the object.

Inside a Solid Sphere

Inside a solid sphere, the gravitational force at a distance $r$ from the center is:

$$ F = G \frac{M_{\text{enc}}m}{r^2} $$

where $M_{\text{enc}}$ is the mass enclosed within a radius $r$. Since the sphere is solid and has uniform density ($\rho$), $M_{\text{enc}}$ can be expressed as:

$$ M_{\text{enc}} = \frac{4}{3} \pi r^3 \rho $$

Thus, the force inside the sphere is:

$$ F = G \frac{\left(\frac{4}{3} \pi r^3 \rho\right)m}{r^2} = \frac{4}{3} \pi G \rho r m $$

Gravitation Force Due to Hollow Spheres

A hollow sphere, also known as a spherical shell, has mass distributed uniformly over its surface. The gravitational force outside a hollow sphere is the same as if all its mass were concentrated at its center. Inside a hollow sphere, the gravitational force is zero everywhere.

Outside a Hollow Sphere

The gravitational force outside a hollow sphere is given by the same formula as for a solid sphere:

$$ F = G \frac{Mm}{r^2} $$

Inside a Hollow Sphere

Inside a hollow sphere, the gravitational force is zero:

$$ F = 0 $$

This is because the contributions to the gravitational force from all parts of the shell cancel each other out.

Comparison Table

Property	Solid Sphere	Hollow Sphere
Mass Distribution	Uniform throughout volume	Uniform over surface
Gravitational Force Outside	$F = G \frac{Mm}{r^2}$	$F = G \frac{Mm}{r^2}$
Gravitational Force Inside	Varies linearly with $r$, zero at center	Zero everywhere
Formula Inside	$F = \frac{4}{3} \pi G \rho r m$	$F = 0$

Examples

Example 1: Gravitational Force Outside a Planet

Consider a planet with mass $M$ and radius $R$. If an object of mass $m$ is located at a distance $r$ from the center of the planet ($r > R$), the gravitational force it experiences is:

$$ F = G \frac{Mm}{r^2} $$

Example 2: Gravitational Force Inside a Planet

If the object is inside the planet at a distance $r$ from the center ($r < R$), and the planet is a solid sphere with uniform density, the gravitational force is:

$$ F = \frac{4}{3} \pi G \rho r m $$

Example 3: Gravitational Force Inside a Hollow Shell

If the object is inside a hollow spherical shell, no matter where it is located inside the shell, the gravitational force it experiences is zero:

$$ F = 0 $$

Understanding the gravitational forces due to hollow and solid spheres is crucial for solving problems related to planetary physics, astrophysics, and other areas where spherical symmetry is a good approximation.