Oscillations and Gravitation
AdvancedSimple harmonic motion arises whenever a restoring force is proportional to displacement: F = -kx for springs, leading to x(t) = A cos(omega t + phi) with omega = sqrt(k/m). The calculus connection is direct: SHM satisfies d^2x/dt^2 = -omega^2 x.
Energy oscillates between kinetic and potential with total E = (1/2)kA^2 constant. Pendulums approximate SHM for small angles with T = 2 pi sqrt(L/g). Gravitation follows an inverse-square law F = GMm/r^2. Gravitational potential energy U = -GMm/r leads to orbital mechanics: circular orbit speed v = sqrt(GM/r), and total orbital energy E = -GMm/(2r).
Kepler laws follow from angular momentum conservation and the inverse-square force.
Practice a little. See where you stand.
Quiz
Reveal what you know — and what needs work
Adaptive Learn
Responds to how you reason, with real-time hints
Flashcards
Build recall through spaced, active review
Cheat Sheet
The essentials at a glance — exam-ready
Glossary
Master the vocabulary that unlocks understanding
Learning Roadmap
A structured path from foundations to mastery
Book
Deep-dive guide with worked examples
Role-play
Think like an expert — no grading
Key Concepts
One concept at a time.
Explore your way
Choose a different way to engage with this topic — no grading, just richer thinking.
Explore your way — choose one:
Curriculum alignment— Standards-aligned
Grade level
Learning objectives
- •Solve the SHM differential equation and interpret amplitude, frequency, and phase
- •Apply energy conservation to oscillating systems
- •Analyze simple and physical pendulums using small-angle approximations
- •Apply Newton law of gravitation and compute gravitational field and potential
- •Derive circular orbit parameters from gravitational force balance
- •Apply Kepler laws to planetary and satellite motion
Recommended Resources
This page contains affiliate links. We may earn a commission at no extra cost to you.
Books
Classical Mechanics
by John R. Taylor
An Introduction to Mechanics
by Daniel Kleppner and Robert Kolenkow
Courses
Related Topics
Physics
Physics is the study of matter, energy, and the fundamental forces of nature, encompassing everything from the motion of everyday objects to the behavior of subatomic particles and the structure of the cosmos.
Newton's Laws
Explore the three fundamental laws governing force, mass, and acceleration that underpin all of classical mechanics.
Calculus-Based Kinematics
Apply derivatives and integrals to describe motion with calculus-based precision.
Energy, Work, and Power
Energy, work, and power describe how forces transfer and transform energy, connecting force and displacement to kinetic, potential, and thermal energy through conservation laws and the work-energy theorem.
Rotational Mechanics
Apply calculus to torque, moment of inertia, angular momentum, and rolling motion.
Conservation of Momentum
Conservation of momentum states that the total momentum of a closed system remains constant through any interaction, enabling analysis of collisions, explosions, and recoil using vector addition of mass-velocity products.