PU-101 · Newtonian Mechanics
Starting from Newton's three laws and the notion of an inertial frame, the unit builds the full deductive machinery of classical particle and rigid-body dynamics, showing how the conservation laws of energy, momentum, and angular momentum follow from force law and symmetry. It culminates in the central-force solution of the Kepler problem and in d'Alembert's principle, which reformulates the whole edifice as the Euler-Lagrange equations and hands the student off to analytical mechanics.
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Derivations homed in this unit
Galilean Invariance of Newton's Laws
Newton's second law is form-invariant under Galilean boosts, translations, and rotations, which is what singles out the inertial frames.
The Work-Energy Theorem
Integrating F = ma along a trajectory shows the work done by the net force equals the change in kinetic energy.
Conservative Forces and Potential Energy
A force field is path-independent iff it is curl-free iff it is the negative gradient of a scalar potential energy.
Conservation of Mechanical Energy
For systems acted on only by conservative forces the sum of kinetic and potential energy is a constant of the motion.
Momentum Conservation from Newton's Third Law
Pairwise internal forces cancel, so the total linear momentum of an isolated system of particles is conserved.
The Center-of-Mass Theorem
The center of mass of a system accelerates as though the total external force acted on the total mass concentrated there.
Angular Momentum, Torque, and Central Forces
The time derivative of angular momentum equals the net torque, so angular momentum is conserved under central forces.
The Two-Body Problem and Reduced Mass
Separating center-of-mass and relative coordinates reduces the two-body central problem to one particle of reduced mass in a fixed potential.
Effective Potential and the Orbit Equation
Angular-momentum conservation collapses a central-force problem to one-dimensional radial motion in an effective potential, yielding the Binet orbit equation.
Kepler's Three Laws from Gravity
Solving the inverse-square orbit equation gives elliptical orbits, the equal-areas law, and the period-semimajor-axis relation.
Small Oscillations and Simple Harmonic Motion
Linearizing any smooth potential about a stable equilibrium yields the harmonic-oscillator equation and its characteristic frequency.
Damped, Driven Oscillators and Resonance
Solving the driven damped oscillator gives the steady-state amplitude, phase lag, and the resonance peak with its quality factor.
Coupled Oscillators and Normal Modes
Diagonalizing the coupled linearized equations of motion gives normal modes as eigenvectors and normal frequencies as eigenvalues.
The Tsiolkovsky Rocket Equation
Applying momentum conservation to a variable-mass body gives the logarithmic relation between velocity gain, exhaust speed, and mass ratio.
The Moment-of-Inertia Tensor
The angular momentum of a rigid body is a linear map of its angular velocity, defining the symmetric inertia tensor and its principal axes.
Euler's Equations and Torque-Free Precession
Writing angular-momentum balance in the body frame yields Euler's equations and predicts free precession and the stability of rotation axes.
Rotating Frames: Centrifugal and Coriolis Forces
Transforming Newton's law into a rotating frame produces the centrifugal, Coriolis, and Euler fictitious forces.
Euler-Lagrange Equations from d'Alembert
d'Alembert's principle in generalized coordinates converts Newton's equations into the Euler-Lagrange equations for the Lagrangian.
Projectile Motion from Constant-Acceleration Kinematics
Integrating constant acceleration gives the kinematic equations; in uniform gravity they make a parabola with range v0^2 sin 2theta / g.
Uniform Circular Motion and Centripetal Acceleration
Differentiating a circular path twice gives an inward acceleration v^2/r, so circular motion needs a net inward force m v^2/r.
Friction: the Coulomb Model and the Inclined Plane
The Amontons-Coulomb laws on an incline: rest while tan theta <= mu_s, slide with a = g(sin theta - mu_k cos theta), friction work lost as heat.
Gravitational Field, Potential, Circular Orbits and Escape Velocity
From the inverse-square law to the potential -GM/r, circular orbit speed sqrt(GM/r), Kepler's third law and escape speed sqrt(2GM/R).