The point mass has done its job
Almost every problem in Volume I secretly replaced a real object with a single point carrying all its mass: a block sliding down an incline, a planet tracing an orbit, a projectile arcing through the air. That trick works whenever an object's size and orientation do not matter. But a wrench tossed across a workshop tumbles end over end; a diving board bends and springs back; the Earth itself wobbles slowly on its axis. None of these can be a point — they have extent, they carry an orientation in space, and they can spin and flex.
This track builds the graduate toolkit for such extended matter, and it rests on two idealizations. The first is the rigid body: a collection of particles whose mutual distances are frozen, so the object can move and turn but never change shape. The second is the deformable continuum: matter smeared into smooth fields that can stretch, shear and vibrate. The first half of this track masters the rotation of rigid bodies; the second half lets them deform.
The great decomposition: translate, then rotate
A remarkable theorem (due to Chasles) says any motion of a rigid body splits cleanly into two parts: the translation of its centre of mass, exactly as if all the mass sat there, plus a rotation about the centre of mass. The centre of mass obeys the same Newtonian law as a point particle — the net force equals total mass times its acceleration — so everything you learned in Volume I still governs that piece. What is genuinely new is the rotation.
This split reaches into the energy too. The total kinetic energy of a rigid body separates into the kinetic energy of the centre of mass plus the rotational kinetic energy about it — König's theorem:
Translational energy of the centre of mass, plus rotational energy carried by the inertia tensor \mathbf{I} acting on the angular velocity \boldsymbol{\omega}.
That object \mathbf{I} — the inertia tensor — is the star of the next guide. For now notice it is not a single number: rotation about different axes costs different amounts of energy, and \mathbf{I} encodes all of them at once.
Angular velocity becomes a vector
The engine of rotation is the angular velocity \boldsymbol{\omega}. Point your right thumb along the rotation axis with the fingers curling in the spin direction: that thumb is the direction of \boldsymbol{\omega}, and its length is the spin rate in radians per second. Encoding rotation as one vector — direction for the axis, magnitude for the rate — is what makes the whole formalism possible.
What makes a body 'rigid' is that every particle shares the same \boldsymbol{\omega} at each instant. The velocity of a particle at position \mathbf{r} (measured from the centre of mass) is then fixed entirely by that one vector:
The rigid-body velocity field: one translation plus one rotation reproduces the motion of every particle in the body.
Two idealizations, and a puzzle to keep
Here is a puzzle to carry with you. Spin a top and stand it on its point: instead of toppling, it leans over and its axis sweeps slowly around a cone — it precesses. A spinning bicycle wheel held by one end of its axle does the same, hanging in mid-air as if gravity had been switched off. By guide 3 you will predict exactly how fast it precesses.
The path ahead: guide 2 builds the inertia tensor and its principal axes — the object that says how a body resists being spun. Guide 3 turns that into Euler's equations and cracks the top and the gyroscope. Then guides 4 and 5 let the body deform: the strain and stress tensors, Hooke's law in three dimensions, and the elastic waves that carry sound through steel and earthquakes through the planet.