Study Notes: Newton's Universal Law of Gravitation

General Principles of Gravitation

• Fundamental Definition: Gravity is a universal force of attraction. Essentially, anything that possesses mass exerts a gravitational pull.

• Planetary Examples of Gravity:     * The Earth and Moon: The Moon is held in its orbit around the Earth by the Earth's gravity.     * The Earth and Sun: The Earth maintains its orbit around the Sun due to the Sun's gravitational force.

• Hypothetical Scenarios of Local Gravity:     * If an individual were positioned in a location in space far enough from any influential mass (like a planet or star), their own mass would exert a noticeable gravitational pull.     * For example, holding an apple in such a void would result in the apple orbiting the person, or more accurately, the two objects orbiting their common center of mass. This occurs because the person's gravity would overcome the negligible external gravity present in the vacuum.

• Earth's Local Acceleration: The gravitational acceleration on Earth's surface is approximately $9.8\,m/s^2$.

• Center of Gravity:     * Earth: The center of gravity for the planet is located at the center of the Earth. Consequently, objects are pulled toward this center.     * Humans: For humans, the center of gravity is located near the solar plexus, often described as the belly button area.

Newton's Third Law and Gravitation

• Equal and Opposite Reactions: According to Newton's Third Law, for every action, there is an equal and opposite reaction.

• Downward vs. Upward Force: If the Earth's gravity exerts an action force that pulls a set of keys toward the ground, the reaction force is the keys pulling the Earth upward.

• Planetary Acceleration:     * Technically, the Earth does accelerate toward the keys as they fall. However, because the mass of the Earth is so immense, this acceleration is not measurable or observable by any practical means.     * Hypothetical Mass Movement: A common inquiry is whether the Earth would move if every person on one side of the planet jumped simultaneously. The consensus is that it would not produce a discernible change due to the Earth's vast mass relative to the population.

Newton's Universal Law of Gravitation

• The Equation: The force of gravity between two objects is calculated using the formula:     $F = G \frac{m_1 \times m_2}{r^2}$     Where:     * $F$ is the gravitational force.     * $G$ is the Universal Gravitational Constant.     * $m_1$ and $m_2$ are the masses of the two objects.     * $r$ is the distance (or radius) between the centers of the two objects.

• The Gravitational Constant ($G$): The value of the universal gravitational constant is $6.67 \times 10^{-11}\,N \cdot m^2/kg^2$.

• Comparison with Coulomb's Constant ($k$):     * The structure of the gravitational law is strikingly similar to Coulomb's Law for electric force: $F_e = k \frac{q_1 \times q_2}{d^2}$.     * In Coulomb's Law, $k = 9 \times 10^9\,N \cdot m^2/C^2$.     * The constant $k$ is a massive number ($9\,billion$) used to magnify the forces of infinitely small charges (like electrons) into measurable scales.     * Conversely, $G$ is an extremely small number because it is used to calculate forces between objects with enormous masses, such as planets, where 10 to the power of 24 is a common scale.

Scaling Gravitational Forces and Mass

• Mass of the Earth ($M_E$): The mass of the Earth is approximately $5.9 \times 10^{24}\,kg$.

• Interactions with Small Objects:     * Personal mass (e.g., $80\,kg$) and everyday objects (e.g., a $1\,kg$ coffee mug) exert gravity on each other.     * However, if you are $10\,meters$ apart, the force would be roughly $4.8 \times 10^{-10}\,N$. This is a force with 10 zeros in front of it, making it entirely unmeasurable.     * We cannot move mugs toward us with our gravity because the Earth's gravity is billions of times stronger and dominates the interaction.

• Impact of the Constant $G$: When multiplying the gravitational constant ($10^{-11}$) by a planetary mass ($10^{24}$), the resulting power of 10 is around $10^{13}\,Newtons$. This scale makes planetary gravity significant and measurable.

Orbital Mechanics and Escape Velocity

• The Distance Factor: Gravity follows the inverse square law. As distance increases, the gravitational pull weakens significantly.

• Pluto vs. Mercury:     * Pluto is so far from the Sun that its orbital velocity is slow, and it has not even completed a full orbit around the Sun since its discovery.     * Mercury is much closer to the sun; because the denominator ($r^2$) is smaller, the gravitational force is much larger, requiring Mercury to orbit the Sun very quickly (roughly every $30$ to $88\,days$).

• Orbital Speed: To achieve a stable orbit around Earth and avoid falling back into the atmosphere, an object must travel at approximately $14,000$ to $16,000\,mph$.

• Escape Velocity: To leave Earth's gravity entirely (e.g., to travel to Mars or into interstellar space), an object must reach escape velocity.     * To get past the Moon ($240,000\,miles$ away) and enter deep space, an object must travel roughly $22,000\,mph$.

• Satellite Motion: Satellites are technically falling in a straight line, but because they are moving so fast, they constantly fall around the curvature of the Earth.

Earth's Curvature and Physical Constants

• Specific Curvature Rate: For every $8,000\,meters$ ($8\,km$) traveled horizontally along the Earth, the surface drops by $5\,meters$ vertically below the horizon.

• Visual Proof regarding the Horizon:     * When viewing large ships on the shoreline, you may only see the top half or third because the ship is so far out that the Earth's curvature has dropped it below your line of sight.     * From the Eastern Coast of Lake Michigan (e.g., New Buffalo, Saint Joseph), one can sometimes see the top third of the Chicago skyline across the water on a clear day, despite being approximately $30$ to $50\,miles$ away.

Comparative Forces and Nature of Gravity

• Gravity vs. Electrostatic Force: Protons and electrons have mass and thus exert gravity, but the force is negligible. In atomic structures, the electrostatic force (attraction/repulsive) and nuclear forces (strong/weak forces) are the primary drivers of movement.

• Non-Contact Force: Unlike conduction, which requires heat transfer through direct contact, gravity is a field force. Objects do not have to touch to exert a gravitational influence. This is similar to radiation, which travels through empty space.

• The Smell of Space: Astronauts on spacewalks have reported a distinct metallic or iron-like taste and smell in the back of their throats after being exposed to the vacuum environment of space.

Mathematical Problem Solving Examples

• Example 1: Force between two objects     * Masses: $4,500\,kg$ each.     * Distance: $2\,m$.     * Calculation: $F = (6.67 \times 10^{-11}) \frac{4,500 \times 4,500}{2^2} = 3.38 \times 10^{-4}\,N$.

• Example 2: Force on a ball near Earth     * Mass of ball: $10\,kg$.     * Radius of Earth: $6.37 \times 10^6\,m$.     * Calculation: $F = (6.67 \times 10^{-11}) \frac{10 \times 5.98 \times 10^{24}}{(6.37 \times 10^6)^2} \approx 98.3\,N$. (Note: This is consistent with the weight calculation $W = mg$, where $10 \times 9.8 = 98$).

• Example 3: Solving for distance (Radius)     * To find the distanceized between two objects based on force, rearrange the formula: $r = \sqrt{\frac{G \times m_1 \times m_2}{F}}$.     * Scenario: Two students with masses of $50\,kg$ and $60\,kg$ have a gravitational force of $3.2 \times 10^{-8}\,N$ between them.     * Result: The distance between them is approximately $2.5\,m$.

Inverse Square Proportionality

• Doubling Distance: If you double the distance ($2 \times r$) between two objects, the gravitational force drops by a factor of 4 ($1/2^2$). It becomes $\frac{1}{4}$ of the original force.

• Halving Distance: If you cut the distance in half ($1/2 \times r$), the force increases by a factor of 4.

• Hypothetical Solar Change: If Earth were twice as close to the Sun, gravity would be four times stronger. A person weighing $150\,lbs$ would weigh $600\,lbs$, making movement extremely difficult.

Questions & Discussion

• Q: Can two objects exert a gravitational force that causes them to repel?     * A: No. Unlike electricity or magnetism, which have attractive and repulsive components, gravity is exclusively attractive.

• Q: Does a satellite exert a force on the Earth?     * A: Yes. Per Newton's Third Law, if the Earth pulls the satellite, the satellite pulls the Earth with an equal force.

• Personal Anecdote (The University Experience):     * The instructor shared a story about a theoretical geometry/calculus professor who also worked at a cyclotron (a miniature particle accelerator like CERN).     * The professor was unapproachable during office hours because he was "working on his book" and treated undergraduate students as if they were unintelligent.     * The professor assigned a paper on the implications of the speed of light being reduced to $30\,m/s$. This would cause noticeable time dilation during flight, making passengers younger upon landing.

• Student Interaction: A student discussed receiving an $86\%$ on a recent test. The instructor reassured the student that an $86\%$ is a very good score, implying they understood the vast majority of the material.