Lesson Notes on Newton's Law of Universal Gravitation
Definition of Newton's Law of Universal Gravitation
Newton's Law of Universal Gravitation states that every point mass attracts every other point mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is given by:
\[ F = G \frac{m_1 \times m_2}{r^2} \]
where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant (\(6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2\)),
- \( m_1 \) and \( m_2 \) are the masses of the objects,
- \( r \) is the distance between the centers of the two masses.
Concept Application
If the mass of one object is doubled (let’s call it \( m_1 \)) and the distance \( r \) between the two objects is halved, the new gravitational force can be expressed as:
\[ F' = G \frac{(2m_1) m_2}{(0.5r)^2} \]
Simplifying the new equation:
\[ F' = G \frac{2m_1 m_2}{0.25r^2} = 8 G \frac{m_1 m_2}{r^2} = 8F \]
Thus, the gravitational force increases by a factor of 8.
Calculation Problem
To calculate the gravitational force between two 5 kg masses that are 2 meters apart:
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Use the formula: \[ F = G \frac{m_1 \times m_2}{r^2} \]
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Substitute the known values:
- \( G = 6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2 \),
- \( m_1 = 5 , \text{kg} \),
- \( m_2 = 5 , \text{kg} \),
- \( r = 2 , \text{m} \).
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Calculate: \[ F = 6.674 \times 10^{-11} \frac{5 \times 5}{2^2} \] \[ F = 6.674 \times 10^{-11} \frac{25}{4} \] \[ F = 6.674 \times 10^{-11} \times 6.25 \] \[ F \approx 4.17 \times 10^{-10} , \text{N} \]
True or False Statement
False: The gravitational force between two objects decreases as the distance between them increases. According to Newton's Law, force is inversely proportional to the square of the distance (i.e., if distance \( r \) increases, \( F \) decreases).
Identification of Factors
Two factors that affect the strength of the gravitational force between two objects are:
- The masses of the objects (greater mass results in a stronger gravitational pull).
- The distance between the centers of the two objects (greater distance results in a weaker gravitational pull).
Real-World Application
Newton's Law of Universal Gravitation helps us understand the orbits of planets around the Sun by providing a framework for predicting how gravitational forces act between celestial bodies. The gravitational pull from the Sun keeps the planets in orbit, while the planets' own momentum moves them forward in their elliptical paths. This mutual interaction explains both the stability and the dynamics of planetary orbits.
These principles are also used in satellite technology and space exploration to calculate trajectories and positions in space.