Physics Notes: Motion, Momentum, Energy, Gravity, and Orbits

Motion, Speed, and Acceleration

• History: evolution of ideas about the universe as matter and energy interacting

• Key terms describing motion:

  • Velocity: speed with direction

  • Speed: how far something travels in a given time

  • Acceleration: change in velocity (in magnitude or direction) over time

• Gravity as acceleration:

  • Acceleration due to gravity is g ≈ 9.8 m/s² on Earth

  • Gravity accelerates all masses equally, independent of their mass (in the classical limit)

• Speed vs velocity:

  • Speed = how far in a time interval

  • Velocity = speed with a specified direction

• Momentum:

  • Definition: $\mathbf{p} = m \mathbf{v}$

  • Momentum can only be changed by applying a force

  • In collisions, momentum is transferred between objects

• Net force:

  • Net force = the combined effect of all individual forces on an object: $\mathbf{F}<em>{\text{net}} = \sum</em>i \mathbf{F}_i$

  • A change in momentum occurs when the net force is not zero: $\Delta \mathbf{p} = \mathbf{F}_{\text{net}} \Delta t$

  • Nonzero net force implies acceleration: $\mathbf{a} = \frac{\mathbf{F}_{\text{net}}}{m}$

• Motion of planets:

  • Planets experience continuous acceleration because their velocity direction changes as they orbit the Sun

Angular Momentum and Torque

• Angular momentum concept:

  • For a rotating or orbiting body, angular momentum is present even if linear velocity is zero (e.g., an ice skater spinning in place)

  • General form: $\mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{r} \times m \mathbf{v}$

  • In simple circular motion, $L = m v r$

• Torque and changes:

  • Torque changes angular momentum: $\boldsymbol{\tau} = \frac{d\mathbf{L}}{dt}$

  • Torque depends on the magnitude and point of application of the force

• Earth’s angular momentum:

  • Rotational angular momentum from Earth’s spin and orbital angular momentum from its orbit around the Sun

Mass vs Weight

• Mass:

  • Amount of matter in an object

• Weight (apparent weight):

  • The force a scale measures when you stand on it; depends on mass and external forces (e.g., gravity, normal forces, etc.)

• Free fall:

  • Falling under gravity with negligible air resistance implies weightlessness in free fall

• Satellites in orbit:

  • Stay in orbit because they are continually falling around Earth without hitting it

Newton’s Revolution in Physics and Basic Laws

• Historical shift:

  • Aristotelian/geocentric views vs Galileo’s observations

  • Newton established laws of motion and gravity, explored light, and helped develop calculus

• Three Newtonian laws (as presented):

  • 1st law (Inertia): An object at constant velocity stays in motion unless acted upon by a net external force; an object at rest remains at rest unless acted upon by a net external force

  • 2nd law (Force and acceleration): $\mathbf{F} = m \mathbf{a}$ (or, more generally, $\mathbf{F} = \frac{d\mathbf{p}}{dt}$ when mass varies)

  • 3rd law (Action–reaction): For every force, there is an equal and opposite reaction force: $\mathbf{F}<em>{12} = -\mathbf{F}</em>{21}$

• Inward forces and orbits:

  • For cars: friction provides the inward force to curve motion; for planets: gravity provides the inward force toward the Sun

Conservation of Momentum

• Momentum conservation in isolated systems:

  • Total momentum before an interaction equals total momentum after: $\sum \mathbf{p}<em>{\text{before}} = \sum \mathbf{p}</em>{\text{after}}$

• Rockets illustrate momentum exchange with expelled exhaust:

  • Forward momentum of rocket is balanced by backward momentum of exhaust, conserving total momentum

• Jumping example:

  • When you push off the Earth with your legs, you gain forward momentum and Earth gains backward momentum; Earth’s huge mass makes the acceleration imperceptible

Conservation of Angular Momentum

• In the absence of external torque, total angular momentum is conserved:

  • Earth’s orbital angular momentum at any point: $L = m v r$ (for simplified circular approximation; general form: $\mathbf{L} = \mathbf{r} \times \mathbf{p}$)

• Why planets orbit: angular momentum conservation implies stability of orbits and relationships between speed and radius

• Rotation of Earth and Moon exchange:

  • Earth’s rotational angular momentum is gradually transferred to the Moon, causing a very slow recession of the Moon

• Ice skater analogy:

  • Pulling arms in increases angular velocity (v) as radius (r) decreases to conserve L

• Astrophysical implications:

  • Spinning disks like galaxies and newborn stars conserve angular momentum during contraction

Where Energy Comes From and Energy Conservation

• Energy conservation principle:

  • Energy cannot be created or destroyed; it is conserved across conversions between forms

• Energy forms:

  • Kinetic energy: $K = \tfrac{1}{2} m v^2$

  • Radiative energy: energy carried by light (photons)

  • Potential energy: energy associated with position in a field (e.g., gravity)

  • Thermal energy: collective random motion of particles; a subset of kinetic energy

• Other energy concepts:

  • Calories and Joules:

  • 1 Calorie (kilocalorie) ≡ 4184 J

  • 1 J = 1 N·m

  • Temperature vs thermal energy:

  • Temperature measures average kinetic energy per particle; thermal energy measures total energy content depending on particle number and density

  • Kelvin scale: absolute temperature scale starting at absolute zero

• Gravitational potential energy and mass–energy equivalence:

  • Gravitational PE depends on mass and height: near Earth, $U_g = m g h$

  • In general, two-body gravity: $U_g(r) = -\frac{G M m}{r}$

  • Mass–energy equivalence: $E = m c^2$

  • Small amounts of mass can contain enormous energy (e.g., nuclear processes)

Gravity and Nuclear Fusion; Big Bang implications

• Gravity and energy origin:

  • Nuclear fusion powers stars, releasing energy that supports light and heat

• Einstein’s insight:

  • Energy can be transformed into mass and vice versa (mass–energy equivalence), underpinning our understanding of nuclear energy and cosmology

• All energy traces back to the Big Bang

• Practical example:

  • Very small mass can yield large energy release in appropriate processes

Gravity: Strength and Mathematical Formulation

• Universal law of gravitation (Newton):
  1) Every mass attracts every other mass via gravity
  2) Gravitational force is proportional to the product of the masses: $F<em>g \propto M m$ 3) Inverse-square law: force falls as the square of the distance between centers: $F</em>g = G \frac{M m}{d^2}$

• Gravitational constant: $G \approx 6.67 \times 10^{-11}\ \mathrm{m^3 \ kg^{-1} \ s^{-2}}$

From Kepler to Newton: Gravity and Orbits

• Kepler’s laws were empirical; Newton showed why they hold by deriving elliptical orbits from the inverse-square law

• Key implications:

  • A planet moves faster when closer to the Sun (conservation of angular momentum)

  • Average orbital speed is slower for planets with larger average orbital distance

  • Kepler’s third law: $p^2 = a^3$ in appropriate units

• Newton’s extension to two bodies:

  • For a small object orbiting a massive one, the orbital period depends on distance and the combined mass: $p^2 = \frac{4\pi^2}{G (M+m)} a^3$

• Center of mass (barycenter):

  • Two bodies orbit their common center of mass; for Sun–Earth, the barycenter is inside the Sun

• Applications:

  • Masses of distant objects can be inferred from orbital period and distance of a companion (e.g., moons around planets)

Orbits and Energy: How Orbits Change

• Orbital energy is conserved in the absence of external work:

  • Total orbital energy: $E = K + U_g = \tfrac{1}{2} m v^2 - \frac{G M m}{r}$

  • As r and v change along the orbit, K and U_g vary, but E remains constant

• Orbital changes via energy exchange:

  • Gravitational encounters (e.g., a spacecraft gaining energy by passing near a planet, like New Horizons with Jupiter)

  • Friction or drag can remove orbital energy (e.g., atmospheric drag on satellites)

• Escape scenarios:

  • If enough energy is added, an object can achieve an unbound trajectory (escape velocity)

• Escape velocity:

  • $v_{\text{esc}} = \sqrt{\frac{2 G M}{r}}$

  • For Earth’s surface, about 11.2 km/s (roughly 40,000 km/h)

  • Escape velocity does not depend on the escaping object’s mass; it depends on the distance from the attracting mass

Tides and Tidal Forces

• Tides arise from gravitational gradient:

  • Moon’s gravity is stronger on the near side of Earth than on the far side, creating tidal bulges

  • This stretching force acts on the entire Earth–Moon line

• Tidal cycle on Earth:

  • Approximately 24 hours and 50 minutes between high tides

  • High tide roughly every 12 hours 25 minutes

  • Tide height and timing depend on latitude, coast geometry, depth, and channels

• Sun’s tidal influence:

  • The Sun’s tidal force is about half the Moon’s on Earth despite the mass difference because the Sun is much further away

• Tidal friction:

  • The bulges are pulled back by Earth's gravity, slowing Earth's rotation and placing energy into the Moon's orbit, causing the Moon to recede slowly

• Synchronous rotation:

  • The Moon always presents the same face to Earth due to tidal locking; a common outcome of tidal interactions in many systems, overtime Earth took rotational angular momentum from the Moon until the Moon stopped rotating

• Tidal deformation can alter shapes and may trigger geological activity (e.g., volcanism) in some bodies

• Long-term consequence:

  • Earth's rotation slows by about one second per ~50,000 years due to tidal friction; the Moon gains angular momentum and moves away

Why Objects Fall at the Same Rate (Equivalence Principle at Earth)

• The equivalence of inertial and gravitational mass:

  • Gravitational acceleration a = F/m = G M / d^2, which is independent of the mass m of the falling object

• At Earth’s surface, this independence yields the familiar acceleration: $g \approx 9.8\ \mathrm{m\,s^{-2}}$

• This is why all objects fall with the same acceleration in a uniform gravitational field, ignoring air resistance

• Practical check:

  • Fg=G((Mearth x M rock)/d^2) where Mearth and Mrock are mass, d is distance from the center of Earth to the center of the rock(6400 km if not far from Earth)

  • If plugged into Newton’s second law:

    G((Mearth x M rock)/d^2)=Mrock x arock

    G(Mearth/d^2) = arock, therefore the acceleration of the rock doesn’t depend on its mass

    Bc the formula applies on earth, a of anything is g, so g=G(Mearth/Rearth^2), plug in Mearth, Rearth, and G values and you get 9.8m/s

Useful Numerical/Conceptual References

• Energy units and conversions:

  • 1 Calorie (kcal) = 4184 J

  • 1 Joule (J) = 1 N·m

• Important constants:

  • Gravitational constant: $G \approx 6.67 \times 10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$

• Inverse-square law emphasizes that doubling distance reduces gravity by a factor of four

• All energy ultimately traces back to cosmological origins (Big Bang) with processes in stars (nuclear fusion) and fundamental physics (mass–energy equivalence)

Connections and Real-World Relevance

• Linking motion, forces, and energy allows prediction of everyday phenomena (car turning, falling objects) and celestial dynamics (orbits, tides)

• Conservation laws (momentum, angular momentum, energy) explain why systems behave predictably without requiring continuous inputs of energy or force

• Understanding tides explains long-term changes in Earth’s rotation rate and Moon’s orbit, as well as potential geological activity on moons and planets

• Kepler’s empirical laws become explained through Newton’s law of gravity, unifying planetary motion with universal gravitation

• The equivalence principle underpins satellite experiments, free-fall physics, and general relativity concepts in more advanced studies

Summary of Key Equations (for quick review)

• Momentum: $\mathbf{p} = m \mathbf{v}$

• Net impulse: $\Delta \mathbf{p} = \mathbf{F}_{\text{net}} \Delta t$

• Newton’s 2nd law (constant mass): $\mathbf{F} = m \mathbf{a} = \frac{d\mathbf{p}}{dt}$

• Angular momentum (general): $\mathbf{L} = \mathbf{r} \times \mathbf{p}$

• Torque: $\boldsymbol{\tau} = \frac{d \mathbf{L}}{dt}$

• Kinetic energy: $K = \tfrac{1}{2} m v^2$

• Gravitational potential energy (near Earth): $U_g = m g h$

• Gravitational potential energy (general two-body): $U_g(r) = -\frac{G M m}{r}$

• Mass–energy equivalence: $E = m c^2$

• Gravitational force: $F_g = G \frac{M m}{d^2}$

• Newton’s extended Kepler: $p^2 = \frac{4\pi^2}{G (M+m)} a^3$

• Orbital energy: $E = K + U_g = \tfrac{1}{2} m v^2 - \frac{G M m}{r}$

• Escape velocity: $v_{\text{esc}} = \sqrt{\frac{2 G M}{r}}$

• Tidal acceleration (order of magnitude): $a_{\text{tidal}} \sim \frac{2 G M R}{d^3}$

• Tidal cycle (Earth): ~24 h 50 m; high tide ~ every 12 h 25 m

• Distance-redshift implications and GER concepts are beyond this scope but part of broader gravity-energy discussion