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Classical Mechanics

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1. Scope and core ideas

Classical mechanics describes the motion of bodies under the action of forces, along with the conservation laws that govern that motion.

It is the foundation for:

  • particle dynamics

  • rigid-body motion

  • vibrations and waves

  • orbital motion

  • mechanical engineering analysis

The core quantities are:

  • position

  • velocity

  • acceleration

  • force

  • momentum

  • energy

  • angular momentum

Modeling assumptions

Most introductory problems rely on simplifying assumptions such as:

  • bodies can be treated as particles

  • objects may be rigid

  • motion occurs in an inertial frame

  • forces can be idealized as weight, normal force, tension, spring force, or drag

Knowing which assumptions are valid is often more important than algebra.

2. Kinematics in one and three dimensions

Kinematics describes motion without explaining its cause.

Position, velocity, and acceleration

In one dimension:

$$ v = \frac{dx}{dt} $$
$$ a = \frac{dv}{dt} = \frac{d^2x}{dt^2} $$

In vector form:

$$ \mathbf{v} = \frac{d\mathbf{r}}{dt} $$
$$ \mathbf{a} = \frac{d\mathbf{v}}{dt} $$

Constant-acceleration relations

For constant acceleration in one dimension:

$$ v = v_0 + at $$
$$ x = x_0 + v_0 t + \frac{1}{2}at^2 $$
$$ v^2 = v_0^2 + 2a(x - x_0) $$
$$ x - x_0 = \frac{1}{2}(v + v_0)t $$

These are valid only when acceleration is constant.

Projectile motion

For ideal projectile motion with no air resistance:

  • horizontal acceleration is zero

  • vertical acceleration is $-g$

So:

$$ x(t) = x_0 + v_{0x}t $$
$$ y(t) = y_0 + v_{0y}t - \frac{1}{2}gt^2 $$

The trajectory is parabolic.

Relative motion

If frame $B$ moves relative to frame $A$, then:

$$ \mathbf{r}_{P/A} = \mathbf{r}_{P/B} + \mathbf{r}_{B/A} $$

Velocities and accelerations follow by differentiating with respect to time.

This is essential in moving-platform and rotating-frame problems.

3. Newton's laws and free-body diagrams

Newton's laws connect force to motion.

Newton's first law

A body remains at rest or moves with constant velocity unless acted on by a net external force.

This defines an inertial frame.

Newton's second law

For a particle:

$$ \sum \mathbf{F} = m\mathbf{a} $$

In components:

$$ \sum F_x = ma_x,\quad \sum F_y = ma_y,\quad \sum F_z = ma_z $$

Newton's third law

For every action there is an equal and opposite reaction.

If body A exerts a force on body B, then body B exerts an equal-magnitude opposite-direction force on body A.

Free-body diagrams

A free-body diagram is the main setup tool in dynamics.

Include only external forces acting on the chosen body.

Typical forces:

  • weight: $\mathbf{W} = m\mathbf{g}$

  • normal force

  • tension

  • spring force: $\mathbf{F} = -k\mathbf{x}$

  • friction

  • drag

Friction

Static friction satisfies:

$$ f_s \le \mu_s N $$

Kinetic friction is approximately:

$$ f_k = \mu_k N $$

Friction opposes relative or impending motion.

Inertial and non-inertial frames

Newton's second law holds in inertial frames.

In accelerating or rotating frames, fictitious forces may be needed to preserve a Newtonian form.

4. Work, energy, and power

Energy methods are often simpler than force balance when displacement is the main unknown.

Work

For a force $\mathbf{F}$ along displacement $d\mathbf{r}$:

$$ dW = \mathbf{F}\cdot d\mathbf{r} $$

So total work is:

$$ W = \int \mathbf{F}\cdot d\mathbf{r} $$

For constant force over displacement $\Delta \mathbf{r}$:

$$ W = \mathbf{F}\cdot \Delta \mathbf{r} $$

Kinetic energy

$$ T = \frac{1}{2}mv^2 $$

Work-energy theorem

Net work equals change in kinetic energy:

$$ W_{\text{net}} = \Delta T $$

Conservative forces and potential energy

A force is conservative if work depends only on initial and final positions.

Examples:

  • gravity

  • ideal spring force

For conservative forces, define potential energy $V$ such that:

$$ \mathbf{F} = -\nabla V $$

In one dimension:

$$ F(x) = -\frac{dV}{dx} $$

Common potentials:

$$ V_g = mgh $$
$$ V_s = \frac{1}{2}kx^2 $$

Conservation of mechanical energy

If only conservative forces act:

$$ T_1 + V_1 = T_2 + V_2 $$

More generally, nonconservative work changes mechanical energy:

$$ T_1 + V_1 + W_{nc} = T_2 + V_2 $$

Power

Power is the rate of doing work:

$$ P = \frac{dW}{dt} $$

For a force:

$$ P = \mathbf{F}\cdot \mathbf{v} $$

5. Momentum, impulse, and collisions

Linear momentum

$$ \mathbf{p} = m\mathbf{v} $$

Newton's second law can be written as:

$$ \sum \mathbf{F} = \frac{d\mathbf{p}}{dt} $$

Impulse

Impulse is the time integral of force:

$$ \mathbf{J} = \int_{t_1}^{t_2} \mathbf{F}\,dt $$

Impulse-momentum relation:

$$ \mathbf{J} = \Delta \mathbf{p} $$

This is especially useful for short-duration forces like impacts.

Conservation of linear momentum

If the net external impulse is negligible, total momentum is conserved:

$$ \sum \mathbf{p}_{\text{before}} = \sum \mathbf{p}_{\text{after}} $$

This is often the best approach for collisions and explosions.

Collisions

Perfectly inelastic collision

Bodies stick together after impact. Momentum is conserved, kinetic energy is not.

Elastic collision

Momentum and kinetic energy are both conserved.

Coefficient of restitution

In one dimension:

$$ e = \frac{\text{relative speed of separation}}{\text{relative speed of approach}} $$

with $0 \le e \le 1$.

Center of mass

For discrete masses:

$$ \mathbf{r}_{cm} = \frac{\sum m_i \mathbf{r}_i}{\sum m_i} $$

The center of mass moves as if all external force were applied there:

$$ \sum \mathbf{F}_{ext} = M\mathbf{a}_{cm} $$

6. Rotation and rigid bodies

Rigid-body motion includes translation and rotation.

Angular variables

Angular displacement:

$$ \theta $$

Angular velocity:

$$ \omega = \frac{d\theta}{dt} $$

Angular acceleration:

$$ \alpha = \frac{d\omega}{dt} $$

Rotational kinematics

For constant angular acceleration:

$$ \omega = \omega_0 + \alpha t $$
$$ \theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2 $$
$$ \omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0) $$

Linear and angular relations

For a point at radius $r$:

$$ v = r\omega $$
$$ a_t = r\alpha $$
$$ a_n = r\omega^2 $$

Tangential acceleration changes speed, normal acceleration changes direction.

Moment of inertia

Moment of inertia measures resistance to angular acceleration:

$$ I = \int r^2\,dm $$

Discrete form:

$$ I = \sum m_i r_i^2 $$

Common results:

  • point mass: $I = mr^2$

  • rod about center: $I = \frac{1}{12}mL^2$

  • rod about end: $I = \frac{1}{3}mL^2$

  • solid disk: $I = \frac{1}{2}mR^2$

  • hoop: $I = mR^2$

Parallel-axis theorem

If $I_{cm}$ is moment of inertia about a center-of-mass axis, then for a parallel axis offset by $d$:

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

Rotational kinetic energy

$$ T_{rot} = \frac{1}{2}I\omega^2 $$

For rolling motion, total kinetic energy is often:

$$ T = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I_{cm}\omega^2 $$

Rolling without slipping

The no-slip condition is:

$$ v_{cm} = R\omega $$

and similarly:

$$ a_{cm} = R\alpha $$

Rolling problems often combine translation, rotation, and static friction.

7. Angular momentum and torque

Torque

Torque is the rotational effect of force:

$$ \boldsymbol{\tau} = \mathbf{r}\times \mathbf{F} $$

Magnitude:

$$ \tau = rF\sin\phi $$

where $\phi$ is the angle between $\mathbf{r}$ and $\mathbf{F}$.

Rotational equation of motion

For a rigid body about a fixed axis:

$$ \sum \tau = I\alpha $$

Angular momentum

For a particle:

$$ \mathbf{L} = \mathbf{r}\times \mathbf{p} $$

For a rigid body about a fixed axis:

$$ L = I\omega $$

Angular impulse-momentum

$$ \int_{t_1}^{t_2}\boldsymbol{\tau}\,dt = \Delta \mathbf{L} $$

Conservation of angular momentum

If the net external torque is zero:

$$ \mathbf{L}_{before} = \mathbf{L}_{after} $$

This is especially useful for:

  • spinning skaters

  • satellites

  • planetary motion

  • rotating collisions

8. Oscillations and simple harmonic motion

Simple harmonic motion

A system exhibits simple harmonic motion when the restoring force is proportional to displacement and directed toward equilibrium:

$$ F = -kx $$

Then:

$$ m\ddot{x} + kx = 0 $$

with solution:

$$ x(t) = A\cos(\omega t + \phi) $$

where

$$ \omega = \sqrt{\frac{k}{m}} $$

Period and frequency

$$ T = \frac{2\pi}{\omega} $$
$$ f = \frac{1}{T} $$

Velocity and acceleration in SHM

If

$$ x(t) = A\cos(\omega t + \phi) $$

then

$$ v(t) = -A\omega\sin(\omega t + \phi) $$
$$ a(t) = -\omega^2 x(t) $$

The acceleration is always directed toward equilibrium.

Energy in SHM

Total energy is constant:

$$ E = \frac{1}{2}kA^2 $$

with exchange between kinetic and potential energy:

$$ T = \frac{1}{2}mv^2,\qquad V = \frac{1}{2}kx^2 $$

Damping and forcing

Real oscillators may include damping and external forcing:

$$ m\ddot{x} + c\dot{x} + kx = F_0\cos(\Omega t) $$

Important behaviors:

  • underdamping: oscillates with decaying amplitude

  • critical damping: fastest return without oscillation

  • overdamping: non-oscillatory and slower

  • resonance: large response near natural frequency

9. Gravitation and central forces

Newton's law of gravitation

Two masses attract with force:

$$ F = G\frac{m_1m_2}{r^2} $$

directed along the line joining them.

Gravitational potential energy

For two masses:

$$ U(r) = -G\frac{m_1m_2}{r} $$

Near Earth, the approximation

$$ U \approx mgh $$

is valid for small height changes.

Circular orbits

For a body of mass $m$ in a circular orbit of radius $r$ around mass $M$:

$$ \frac{mv^2}{r} = G\frac{Mm}{r^2} $$

so

$$ v = \sqrt{\frac{GM}{r}} $$

The orbital period is

$$ T = 2\pi \sqrt{\frac{r^3}{GM}} $$

Central-force intuition

In central-force motion:

  • force points toward or away from a center

  • angular momentum is conserved

  • motion often reduces to an effective one-dimensional radial problem

10. Constraints, equilibrium, and statics

Static equilibrium means no translational or rotational acceleration.

Equilibrium conditions

For a particle:

$$ \sum \mathbf{F} = 0 $$

For a rigid body:

$$ \sum \mathbf{F} = 0 $$
$$ \sum \tau = 0 $$

Typical statics workflow

  1. Draw the free-body diagram.

  2. Choose convenient axes.

  3. Write force-balance equations.

  4. Write torque-balance equations about a smart pivot.

  5. Solve for unknown reactions, tensions, or loads.

Constraints

Constraints reduce the degrees of freedom of a system.

Examples:

  • bead constrained on a wire

  • block on an incline

  • pulley system

  • rigid rod with pinned supports

Constraint forces often do no work in idealized models, but they still matter in force balance.

11. Lagrangian and Hamiltonian ideas

The Lagrangian formulation is a higher-level way to derive equations of motion.

Generalized coordinates

Instead of using Cartesian coordinates directly, choose coordinates $q_i$ that describe the degrees of freedom efficiently.

Examples:

  • pendulum angle $\theta$

  • spring extension $x$

  • orbital radius $r$

Lagrangian

$$ \mathcal{L} = T - V $$

The Euler-Lagrange equation is:

$$ \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q}_i}\right) - \frac{\partial \mathcal{L}}{\partial q_i} = 0 $$

for each generalized coordinate $q_i$.

Why use Lagrange's equations

  • constraints are handled naturally

  • coordinate choice can match the geometry

  • fewer force components may need to be resolved

Canonical momentum

For coordinate $q_i$:

$$ p_i = \frac{\partial \mathcal{L}}{\partial \dot{q}_i} $$

Hamiltonian idea

The Hamiltonian is often the total energy written in terms of coordinates and momenta:

$$ \mathcal{H} = \sum_i p_i \dot{q}_i - \mathcal{L} $$

In many standard mechanical systems,

$$ \mathcal{H} = T + V $$

though this is not universal in all formulations.

12. Problem-solving workflow

Start with the model

Ask:

  • Is the body a particle, rigid body, or system of interacting bodies?

  • Is the goal acceleration, velocity, position, force, energy, or time?

  • Are there constraints, collisions, or rotational effects?

Choose the right tool

  • Use Newton's laws for direct force-acceleration problems.

  • Use energy when forces are conservative or the path is complicated.

  • Use momentum for collisions and short impulses.

  • Use torque and angular momentum for rotation.

  • Use equilibrium equations for static systems.

Common pitfalls

  • Mixing up mass and weight

  • Using energy methods while forgetting nonconservative work

  • Forgetting direction in vector quantities

  • Applying constant-acceleration formulas when acceleration is not constant

  • Using the wrong sign convention for torque or angular variables

  • Treating a rolling object as pure translation

  • Ignoring whether the frame is inertial

Sanity checks

  • Do units match on both sides?

  • Does the result reduce to a known special case?

  • Does the sign make physical sense?

  • Is the answer bounded or physically plausible?

13. Formula summary

Kinematics

$$ v = \frac{dx}{dt}, \qquad a = \frac{dv}{dt} $$
$$ v = v_0 + at $$
$$ x = x_0 + v_0t + \frac{1}{2}at^2 $$

Newtonian dynamics

$$ \sum \mathbf{F} = m\mathbf{a} $$

Work and energy

$$ W = \int \mathbf{F}\cdot d\mathbf{r} $$
$$ T = \frac{1}{2}mv^2 $$
$$ W_{net} = \Delta T $$
$$ T_1 + V_1 + W_{nc} = T_2 + V_2 $$

Momentum

$$ \mathbf{p} = m\mathbf{v} $$
$$ \mathbf{J} = \int \mathbf{F}\,dt = \Delta \mathbf{p} $$

Rotation

$$ \tau = rF\sin\phi $$
$$ \sum \tau = I\alpha $$
$$ L = I\omega $$
$$ \mathbf{L} = \mathbf{r}\times \mathbf{p} $$
$$ T_{rot} = \frac{1}{2}I\omega^2 $$

Spring and SHM

$$ F = -kx $$
$$ \omega = \sqrt{\frac{k}{m}} $$
$$ T = \frac{2\pi}{\omega} $$

Gravitation

$$ F = G\frac{m_1m_2}{r^2} $$
$$ U = -G\frac{m_1m_2}{r} $$

Rolling

$$ v_{cm} = R\omega $$

Lagrangian form

$$ \mathcal{L} = T - V $$
$$ \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q}_i}\right) - \frac{\partial \mathcal{L}}{\partial q_i} = 0 $$

Compact takeaway

Classical mechanics is mostly about choosing the right representation:

  • force balance for instantaneous dynamics

  • energy for path-independent force fields

  • momentum for impacts and isolated systems

  • torque and angular momentum for rotation

  • generalized coordinates for constrained motion

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