🌊College Physics II – Mechanics, Sound, Oscillations, and Waves Unit 13 – Gravitation: Fundamental Force of Attraction

Key Concepts and Definitions

• Gravitation the attractive force between any two objects with mass, proportional to their masses and inversely proportional to the square of the distance between them
• Mass a measure of an object's inertia and its resistance to acceleration, determines the strength of its gravitational field
• Weight the force exerted on an object due to gravity, equal to the object's mass multiplied by the acceleration due to gravity ($F_g = mg$)
• Gravitational constant ($G$) a fundamental physical constant that determines the strength of gravity, equal to $6.67 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2$
  • Used in Newton's law of universal gravitation to calculate the force of gravity between two objects
• Gravitational field the region around a massive object in which another object experiences a gravitational force
  • Represented by field lines that point in the direction of the gravitational force
• Gravitational potential energy the energy an object possesses due to its position in a gravitational field, equal to the work done by gravity to move the object from infinity to its current position ($U = -GMm/r$)

Historical Context and Development

• Ancient Greeks (Aristotle) believed that objects fall towards the Earth because it is their natural place, heavier objects fall faster than lighter ones
• Galileo Galilei challenged Aristotelian ideas, performed experiments with inclined planes and pendulums to study motion and gravity
  • Discovered that all objects fall at the same rate regardless of their mass (neglecting air resistance)
• Isaac Newton developed the law of universal gravitation, explaining the force of gravity between any two objects with mass
  • Published in his famous work "Principia Mathematica" (1687)
• Cavendish experiment (1798) first measurement of the gravitational constant ($G$) using a torsion balance
• Albert Einstein's theory of general relativity (1915) described gravity as a curvature of spacetime caused by the presence of mass and energy
  • Explained phenomena such as the precession of Mercury's orbit and the bending of light by massive objects

Newton's Law of Universal Gravitation

• States that every particle in the universe attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them
• Mathematical formula: $F_g = G \frac{m_1m_2}{r^2}$
  • $F_g$ is the gravitational force between two objects
  • $G$ is the gravitational constant
  • $m_1$ and $m_2$ are the masses of the two objects
  • $r$ is the distance between the centers of the two objects
• The force is always attractive and acts along the line connecting the centers of the two objects
• Applies to all objects in the universe, from subatomic particles to galaxies
• Explains the motion of planets around the Sun, the Moon around the Earth, and the formation of large-scale structures in the universe

Gravitational Field and Potential

• Gravitational field the force per unit mass experienced by a test particle at a given point in space
  • Represented by the vector $\vec{g} = -G \frac{M}{r^2} \hat{r}$, where $M$ is the mass of the source object and $\hat{r}$ is the unit vector pointing from the source to the test particle
• Gravitational field lines imaginary lines that represent the direction and strength of the gravitational field
  • Always point towards the source of the field (the massive object)
  • Closer spacing of field lines indicates a stronger gravitational field
• Gravitational potential energy the energy an object possesses due to its position in a gravitational field
  • Calculated using the formula $U = -G \frac{Mm}{r}$, where $M$ is the mass of the source object, $m$ is the mass of the test particle, and $r$ is the distance between them
  • Negative sign indicates that the gravitational force is attractive, and the potential energy decreases as the objects move closer together
• Escape velocity the minimum speed an object needs to escape the gravitational pull of a massive body (like a planet or star)
  • Calculated using the formula $v_e = \sqrt{\frac{2GM}{r}}$, where $M$ is the mass of the source object and $r$ is the distance from its center

Orbital Motion and Kepler's Laws

• Kepler's laws of planetary motion three empirical laws that describe the motion of planets around the Sun
  • Kepler's first law (law of ellipses) planets orbit the Sun in elliptical paths, with the Sun at one focus of the ellipse
  • Kepler's second law (law of equal areas) a line segment joining a planet and the Sun sweeps out equal areas in equal intervals of time
  • Kepler's third law (law of periods) the square of a planet's orbital period is directly proportional to the cube of the semi-major axis of its orbit
• Orbital velocity the speed at which an object must travel to maintain a stable orbit around a massive body
  • Calculated using the formula $v_o = \sqrt{\frac{GM}{r}}$, where $M$ is the mass of the central body and $r$ is the orbital radius
• Orbital period the time it takes for an object to complete one full orbit around a massive body
  • Calculated using Kepler's third law: $T^2 = \frac{4\pi^2}{GM}a^3$, where $T$ is the orbital period, $M$ is the mass of the central body, and $a$ is the semi-major axis of the orbit
• Geosynchronous orbit an orbit around the Earth with an orbital period equal to one sidereal day (approximately 23 hours, 56 minutes, 4 seconds)
  • Satellites in geosynchronous orbits appear to remain stationary above a fixed point on the Earth's equator

Gravitational Effects on Earth

• Tides the rise and fall of sea levels caused by the gravitational pull of the Moon and Sun on the Earth's oceans
  • Spring tides occur when the Earth, Moon, and Sun are aligned, resulting in higher high tides and lower low tides
  • Neap tides occur when the Moon and Sun are at right angles to the Earth, resulting in lower high tides and higher low tides
• Precession of the equinoxes a slow, conical motion of the Earth's rotational axis caused by the gravitational torque exerted by the Sun and Moon on the Earth's equatorial bulge
  • One complete precession cycle takes approximately 25,800 years
• Gravitational time dilation the slowing of time in the presence of a strong gravitational field, as predicted by Einstein's theory of general relativity
  • Clocks on Earth's surface run slightly slower than clocks at higher altitudes due to the Earth's gravitational field
• Gravitational lensing the bending of light by massive objects, causing the light from distant sources to be distorted or magnified
  • Can be used to detect dark matter and study distant galaxies

Applications in Space Exploration

• Gravity assist a technique used in spaceflight to change a spacecraft's velocity and trajectory by using the gravitational field of a planet or moon
  • Spacecraft can gain or lose velocity depending on the relative motion between the spacecraft and the assisting body
• Lagrange points five positions in an orbital configuration where a small object (like a spacecraft) can maintain a stable position relative to two larger objects (like the Earth and the Moon)
  • L1, L2, and L3 are unstable equilibrium points, while L4 and L5 are stable equilibrium points
  • James Webb Space Telescope (JWST) orbits the Sun-Earth L2 point, approximately 1.5 million kilometers from Earth
• Gravitational wave detection the observation of ripples in spacetime caused by massive cosmic events, such as the merger of two black holes or neutron stars
  • First detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO) in 2015
  • Provides a new way to study the universe and test Einstein's theory of general relativity

Advanced Topics and Current Research

• Dark matter a hypothetical form of matter that does not interact with electromagnetic radiation (light) but has gravitational effects on visible matter
  • Believed to make up approximately 85% of the matter in the universe
  • Candidates include weakly interacting massive particles (WIMPs), axions, and modified gravity theories
• Dark energy a hypothetical form of energy that permeates all of space and causes the expansion of the universe to accelerate
  • Accounts for approximately 68% of the total energy in the observable universe
  • Can be described by the cosmological constant ($\Lambda$) in Einstein's field equations or by scalar fields such as quintessence
• Modified gravity theories alternative theories that attempt to explain gravitational phenomena without invoking dark matter or dark energy
  • Examples include Modified Newtonian Dynamics (MOND), Tensor-Vector-Scalar gravity (TeVeS), and $f(R)$ gravity
  • Aim to reconcile observations on galactic and cosmological scales with the predictions of general relativity
• Quantum gravity the attempt to develop a theory that unifies quantum mechanics and general relativity, describing gravity at the quantum scale
  • Candidates include string theory, loop quantum gravity, and causal dynamical triangulations
  • Seeks to resolve incompatibilities between the two theories and explain the behavior of gravity at extremely high energies and small scales (Planck scale)