Simple Harmonic Motion
x(t) = A cos(omega t + phi), where omega = sqrt(k/m).
Example: 0.5 kg on k=200 N/m spring: omega = 20 rad/s.
Adaptive
Learn Oscillations and Gravitation
Read the notes, then try the practice. It adapts as you go.When you're ready.
Session Length
~18 min
Adaptive Checks
16 questions
Transfer Probes
8
Lesson Notes
Simple 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.
You'll be able to:
- 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
One step at a time.
Interactive Exploration
Adjust the controls and watch the concepts respond in real time.
Key Concepts
SHM Differential Equation
d^2x/dt^2 = -omega^2 x.
Example: x=A cos(omega t) satisfies this ODE.
Energy in SHM
E = (1/2)kA^2 = (1/2)mv^2 + (1/2)kx^2.
Example: A=0.1m, k=200: E=1 J.
Simple Pendulum
T = 2 pi sqrt(L/g) for small angles.
Example: L=1m: T = 2 pi sqrt(1/9.8) = 2.01 s.
Physical Pendulum
T = 2 pi sqrt(I/(mgh)).
Example: Rod about end: T = 2 pi sqrt(2L/3g).
Newton Law of Gravitation
F = GMm/r^2, attractive along line of centers.
Example: Earth-Moon force calculation.
Gravitational Potential Energy
U = -GMm/r. Zero at infinity.
Example: Escape velocity: v = sqrt(2GM/R).
Circular Orbits
v = sqrt(GM/r), T^2 = (4 pi^2/GM)r^3.
Example: LEO at 400 km: v = 7.67 km/s.
Explore your way
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Concept Map
See how the key ideas connect. Nodes color in as you practice.
Worked Example
Walk through a solved problem step-by-step. Try predicting each step before revealing it.
Adaptive Practice
This is guided practice, not just a quiz. Hints and pacing adjust in real time.
Small steps add up.
What you get while practicing:
- Math Lens cues for what to look for and what to ignore.
- Progressive hints (direction, rule, then apply).
- Targeted feedback when a common misconception appears.
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