Newton’s Law of Gravitation
The Universal Law of Gravitation
- Newton's law of gravitation describes the attractive force between any two objects with mass.
- The force is given by: $$F = G \frac{m_1 m_2}{r^2}$$ where:
- $F$ is the gravitational force between the masses (in newtons, N).
- $G$ is the universal gravitational constant, approximately $6.67 \times 10^{-11} \, \text{N m}^2 \ \text{kg}^{-2}$.
- $m_1$ and $m_2$ are the masses of the two objects (in kilograms, kg).
- $r$ the distance between the centers of the two masses (in meters, m).
The gravitational force is always attractive and acts along the line joining the two masses.
Application to Spherical Masses
Newton proved that for spherical bodies with uniform density, the gravitational force can be calculated as if all the mass were concentrated at the center of the sphere.
This simplifies calculations for planets, stars, and other spherical objects.
Calculate the gravitational force between Earth and the Moon, given:
- Mass of Earth: $m_1 = 5.97 \times 10^{24} \, \text{kg}$
- Mass of Moon: $m_2 = 7.35 \times 10^{22} \, \text{kg}$
- Distance between centers: $r = 3.84 \times 10^8 \, \text{m}$
Solution
- Substitute the given values into the formula:
$$F = G \frac{m_1 m_2}{r^2} = (6.67 \times 10^{-11}) \frac{(5.97 \times 10^{24})(7.35 \times 10^{22})}{(3.84 \times 10^8)^2}$$
$$F \approx 1.98 \times 10^{20} \, \text{N}$$
- This is the gravitational force between Earth and the Moon.
Conditions for Point Mass Approximation
Newton’s law of gravitation is derived for point masses, but it can be applied to extended bodies under certain conditions:
- Spherical Symmetry: The body must be spherical and have a uniform mass distribution.
- Distance: The distance between the bodies must be much larger than their sizes.
- For non-spherical bodies, the gravitational force can be complex.
- However, if the body is far enough away, it can often be approximated as a point mass.
Gravitational Field Strength
Definition of Gravitational Field Strength
Gravitational field
A gravitational field is a region of space where a mass experiences a gravitational force.
Gravitation field strength
The gravitational field strength ($g$) at a point is defined as the force per unit mass experienced by a small test mass placed at that point.
Mathematically:
$$g = \frac{F}{m}$$
