Gravitational Force Inside the Earth: Videos & Practice Problems

When calculating gravitational force inside a planet, consider that only the mass beneath you contributes to the force. The gravitational force increases linearly with distance from the center, expressed as F_inside = G(M) m(r) for r<R. At the center, gravitational force is zero, resulting in weightlessness. Understanding these principles is crucial for solving related problems in gravitational physics.

Gravitational Force Inside the Earth

Newton's law of gravity describes the gravitational force between two masses, typically applied when objects are on or above the Earth's surface. However, when considering an object located within the Earth, a different approach is necessary. Inside the Earth, the gravitational force can be understood by concentrating all the mass beneath the object at the center of the Earth. This means that when digging down, the gravitational force depends only on the mass of the Earth that is closer to the center than the object itself, effectively treating the object as if it were on the surface of a smaller, less massive Earth.

The gravitational force experienced by an object inside the Earth can be expressed with the formula:

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

where \( F \) is the gravitational force, \( G \) is the gravitational constant, \( m \) is the mass of the object, \( M_{\text{inside}} \) is the mass of the Earth within the radius \( r \), and \( r \) is the distance from the center of the Earth to the object.

To derive the mass \( M_{\text{inside}} \), we can relate the density of the inner sphere to the overall density of the Earth. Assuming uniform density, we can express the mass of the inner sphere as:

$$ M_{\text{inside}} = \rho \cdot V_{\text{inside}} $$

where \( \rho \) is the density of the Earth and \( V_{\text{inside}} \) is the volume of the inner sphere, given by:

$$ V_{\text{inside}} = \frac{4}{3} \pi r^3 $$

Thus, the mass inside can be expressed as:

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

Substituting this back into the gravitational force equation leads to a simplified expression for gravitational force inside the Earth:

$$ F = \frac{G \cdot m \cdot \rho \cdot \frac{4}{3} \pi r^3}{r^2} $$

After simplification, this results in:

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

This indicates that the gravitational force inside the Earth is directly proportional to the distance \( r \) from the center, meaning that as one moves closer to the center, the gravitational force increases until reaching the surface, where it begins to decrease with distance from the center, following the inverse square law.

At the very center of the Earth, where \( r = 0 \), the gravitational force is zero, resulting in a state of weightlessness due to the symmetrical distribution of mass surrounding the object.

To illustrate this concept, consider a person with a surface weight of 780 Newtons who drills down into the Earth. If their weight at a certain depth is 80% of their surface weight, we can set up the equation:

$$ F_{\text{inside}} = 0.80 \times 780 \, \text{N} = 624 \, \text{N} $$

Using the gravitational force equation for inside the Earth, we can rearrange to find the distance \( r \) at which this weight occurs:

$$ r = \frac{F_{\text{inside}} \cdot r^3}{G \cdot M \cdot m} $$

By calculating the mass of the person from their weight and substituting known values for the radius of the Earth and gravitational constant, we can determine the depth at which their weight is 624 N. This results in a distance \( r \) that can be expressed as a fraction of the Earth's radius, confirming that the weight inside is proportional to the distance from the center.

In summary, understanding how gravitational force varies within the Earth is crucial for solving problems related to objects located beneath the surface, emphasizing the importance of mass distribution and the concept of weightlessness at the Earth's center.

A uniform, solid, 1600.0 kg sphere has a radius of 5.00 m. Find the gravitational force this sphere exerts on a 2.10 kg point mass placed at the following distances from the center of the sphere: (a) 5.10 m, and (b) 2.55 m.

• A. (a) $67.5\times10^{-9}$ N(b) $4.57\times10^{-9}$ N
• B. (a) $8.67\times10^{-9}$ N(b) $67.5\times10^{-9}$ N
• C. (a) $8.67\times10^{-9}$ N(b) $4.57\times10^{-9}$ N
• D. (a) $67.5\times10^{-9}$ N(b) $34.4\times10^{-9}$ N

Answer: (C)

Here’s what students ask on this topic:

What is the gravitational force inside the Earth?

Inside the Earth, the gravitational force is given by the equation $F=\frac{GMmr}{R^3}$, where $G$ is the gravitational constant, $M$ is the mass of the Earth, $m$ is the mass of the object, $r$ is the distance from the center of the Earth, and $R$ is the radius of the Earth. The force increases linearly with distance from the center until it reaches the surface, where it follows the inverse-square law.

How does gravitational force change as you move towards the center of the Earth?

As you move towards the center of the Earth, the gravitational force decreases linearly with distance. This is because only the mass beneath you contributes to the gravitational force. The force is given by $F=\frac{GMmr}{R^3}$. At the very center, the gravitational force is zero, resulting in weightlessness, as the mass surrounding you cancels out.

Why is the gravitational force zero at the center of the Earth?

The gravitational force is zero at the center of the Earth because the mass of the Earth is symmetrically distributed around you. This symmetry causes the gravitational forces from all directions to cancel each other out, resulting in a net force of zero. This concept is derived from the shell theorem, which states that a spherical shell of uniform mass exerts no net gravitational force on an object located inside it.

How do you calculate the gravitational force inside a planet with uniform density?

To calculate the gravitational force inside a planet with uniform density, use the formula $F=\frac{GMmr}{R^3}$, where $G$ is the gravitational constant, $M$ is the mass of the planet, $m$ is the mass of the object, $r$ is the distance from the center, and $R$ is the planet's radius. This formula assumes the planet has a uniform density, allowing the mass inside to be proportional to the volume.

What is the relationship between gravitational force and distance inside the Earth?

Inside the Earth, the gravitational force is directly proportional to the distance from the center. This relationship is expressed as $F=\frac{GMmr}{R^3}$, where $G$ is the gravitational constant, $M$ is the mass of the Earth, $m$ is the mass of the object, $r$ is the distance from the center, and $R$ is the Earth's radius. This linear relationship holds until you reach the surface, where the force follows the inverse-square law.