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Classical Mechanics

Newton's laws, energy, momentum — the physics robots obey.

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Prerequisites

Calculus & Optimization

Why it matters in robotics

Classical mechanics is the foundation beneath everything a robot does physically: every dynamics model, controller, and trajectory rests on Newton's laws, forces, torques, and the conservation of energy and momentum. Free-body diagrams, F=ma and torque-based rotational equations for a link or wheel, how a body's mass distribution affects its rotation, and energy or momentum methods for collisions and impacts are the basic tools. They connect directly to robotics — why mass distribution matters for a spinning rotor, how friction limits the forces a gripper can apply without slipping, and how energy reasoning simplifies working out how a mechanism moves.

What to study

✓Forces, torques, and free-body diagrams: resolve forces, sum moments, and apply $F=ma$ / $\tau=I\alpha$ to multi-body systems with constraints.
✓Work, energy, and momentum: kinetic/potential energy, the work-energy theorem, and conservation of linear and angular momentum (including elastic vs. inelastic collisions).
✓Rotational dynamics and moment of inertia: parallel-axis theorem, the inertia tensor, rolling without slipping, and angular momentum of rigid bodies.
✓Friction, contact, and constraints: static vs. kinetic friction, the friction cone, normal/contact forces, and how holonomic constraints reduce degrees of freedom.

Where to study

Cheatsheet

Symbols used below

F

net force

m

mass

a

linear acceleration

\tau

torque

I

moment of inertia

\alpha

angular acceleration

\omega

angular velocity

p

linear momentum

L

angular momentum

\mu

coefficient of friction

N

normal force

d

distance from a reference axis

Newton's second law (linear & rotational)

$F = ma$

The rotational analog swaps force for torque and mass for moment of inertia:

$\tau = I\alpha$

Work, energy, and momentum

Kinetic energy: $KE = \tfrac{1}{2}mv^2$ (linear), $KE = \tfrac{1}{2}I\omega^2$ (rotational)

Work-energy theorem: net work done equals the change in kinetic energy.

Linear momentum $p = mv$ and angular momentum $L = I\omega$ are each conserved when no external force/torque acts.

Elastic vs. inelastic collisions

Both conserve momentum. Only an elastic collision also conserves kinetic energy.

In a perfectly inelastic collision the bodies stick together afterward, and kinetic energy is lost (converted to heat/deformation) even though momentum is unchanged.

Parallel-axis theorem

$I = I_{cm} + md^2$

Moment of inertia about any axis equals the inertia about a parallel axis through the center of mass, plus $md^2$ for the offset $d$ . This is why a motor's rotor and an off-axis payload contribute very differently to the inertia a joint feels.

Rolling without slipping

$v = \omega r$

The contact point's velocity is zero relative to the ground — linear and angular motion are locked together. Break this constraint (wheel spin or skid) and the relation no longer holds.

Friction cone

Static friction can supply up to $F_{friction} \le \mu N$ before slipping begins.

This caps how much lateral force a gripper or wheel can apply without slip — a gripper squeezing with normal force $N$ can only resist a payload's weight/acceleration up to $\mu N$ before it slides.

Application focus

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