Kepler's Laws and Gravitation

This cheat sheet covers Kepler's Laws of planetary motion, Newton's Law of Universal Gravitation, and the concept of gravitational fields. It explores how these laws describe the motion of celestial bodies and the fundamental force that governs them.

Core Principles

• Kepler's First Law: Planets move in elliptical orbits with the Sun at one focus.
• Kepler's Second Law: A line connecting a planet to the Sun sweeps out equal areas in equal time intervals.
• Kepler's Third Law: The ratio of the cube of the average orbital radius to the square of the period for each revolution is constant ($r^3/T^2 = constant$).
• Newton's Law of Universal Gravitation: The force of attraction between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• Gravitational Field: A region of space where a mass experiences a force due to another mass. The field strength decreases with distance.
• Inverse Square Law: Many physical properties decrease in intensity with the square of the distance from the source.

Action Steps

• Identify the type of field (gravitational, electric, magnetic).
• Determine the relevant formula based on the field and the quantities involved.
• Ensure all units are consistent before calculation.
• Apply the correct right-hand rule (grip or slap) for magnetic fields to determine direction.
• Calculate forces, fields, energy, or motion using the appropriate equations.
• Interpret results, paying attention to signs (attraction/repulsion, direction).

Formulas

• Kepler's Third Law: $r^3/T^2 = constant$
• Newton's Law of Universal Gravitation: $F = G(m_1 m_2)/r^2$
• Gravitational Field Strength: $g = GM/r^2$
• Force due to gravity: $F_g = mg$
• Kinetic Energy: $KE = 1/2 mv^2$
• Work done: $W = Fd$
• Charge: $Q = I \times t$
• Electric Force: $F = qE$
• Electric Field Strength: $E = V/d$
• Work done by electric field: $W = qV$
• Coulomb's Law: $F = kq_1q_2/r^2$
• Electric Field Strength (point charge): $E = kQ/r^2$
• Lorentz Force: $F = qvB$
• Magnetic force on a current-carrying wire: $F = nIlB$
• Torque: $ au = r_{\perp} F$
• Kinetic energy of particle from electron gun: $1/2 mv^2 = qV$
• Radius of particle in magnetic field: $r = mv/qB$

Key Terms

• Ellipse: A closed curve, defined by two focal points, such that the sum of the distances from any point on the curve to the two focal points is constant. Planets orbit the Sun in ellipses.
• Focus: One of the two fixed points used to define an ellipse. The Sun is at a focus of Earth's elliptical orbit.
• Areal Speed: The rate at which a line connecting a planet to the Sun sweeps out area. Kepler's Second Law states this is constant.
• Universal Gravitation: Newton's law describing the attractive force between any two masses.
• Gravitational Field: The region around a mass where another mass would experience a gravitational force.
• Inverse Square Law: A relationship where a quantity is inversely proportional to the square of the distance from a source.
• Charge: A fundamental property of matter that causes it to experience a force when placed in an electric or magnetic field. Measured in Coulombs (C).
• Electric Field: A region around a charged object where another charged object experiences a force.
• Magnetic Field: A region around a magnetic material or a moving electric charge within which the force of magnetism acts.
• Lorentz Force: The force experienced by a charged particle moving in a magnetic field.
• Right-Hand Grip Rule: A rule used to determine the direction of the magnetic field around a current-carrying wire or loop.
• Right-Hand Slap Rule: A rule used to determine the direction of the force on a current-carrying wire or moving charge in a magnetic field.
• Torque: A twisting or turning force that causes rotation.
• Commutator: A component in an electric motor that reverses the direction of the current in the coil to maintain continuous rotation.
• Particle Accelerator: A machine that uses electromagnetic fields to propel charged particles to very high speeds.

Timeline

• 17th Century: Johannes Kepler formulates his three laws of planetary motion.
• Late 17th Century: Isaac Newton develops the law of universal gravitation and his laws of motion.
• 18th Century: Charles-Augustin de Coulomb investigates electrostatic forces, leading to Coulomb's Law.
• 19th Century: Michael Faraday and James Clerk Maxwell develop theories of electromagnetism.
• Early 20th Century: Development of particle accelerators to study subatomic particles.

People

• Johannes Kepler: Astronomer and mathematician who formulated the laws of planetary motion.
• Isaac Newton: Physicist and mathematician who developed the law of universal gravitation.
• Charles-Augustin de Coulomb: Physicist who formulated Coulomb's Law regarding electrostatic forces.
• Michael Faraday: Scientist who discovered electromagnetic induction and laid the groundwork for electromagnetism.
• James Clerk Maxwell: Physicist who unified electricity, magnetism, and light into a single theory.