Physics: Classical Mechanics

Introduction

Classical mechanics describes the motion of macroscopic objects. From falling apples to orbiting planets, the equations of mechanics govern the physical world around us.

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Kinematics: Motion in One Dimension

Position and Displacement

The position of an object is described by $x(t)$, a function of time. Displacement is:

$\Delta x = x_f - x_i$

Velocity

Average velocity:

$\bar{v} = \frac{\Delta x}{\Delta t}$

Instantaneous velocity:

$v = \frac{dx}{dt}$

Acceleration

Average acceleration:

$\bar{a} = \frac{\Delta v}{\Delta t}$

Instantaneous acceleration:

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

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Kinematic Equations

For constant acceleration $a$, we have the four kinematic equations:

$v = v_0 + at$

$x = x_0 + v_0 t + \frac{1}{2}at^2$

$v^2 = v_0^2 + 2a(x - x_0)$

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

where:

• $v_0$ = initial velocity
• $v$ = final velocity
• $a$ = acceleration
• $t$ = time
• $x_0$ = initial position
• $x$ = final position

Interactive: Projectile Motion

The chart below shows position vs. time for an object dropped from rest (no air resistance, $g = 9.8 \, \text{m/s}^2$):

And here is the corresponding velocity vs. time graph ($v = gt$):

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Newton's Laws of Motion

First Law (Inertia)

An object at rest stays at rest, and an object in motion stays in motion with the same speed and direction, unless acted upon by an unbalanced force.

Second Law (Force and Acceleration)

The net force on an object equals its mass times its acceleration:

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

This is the fundamental equation of mechanics. From it, we can derive:

$\vec{a} = \frac{\sum \vec{F}}{m}$

$m = \frac{\sum \vec{F}}{\vec{a}}$

Third Law (Action-Reaction)

For every action, there is an equal and opposite reaction:

$\vec{F}_{12} = -\vec{F}_{21}$

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Types of Forces

Gravitational Force

Newton's law of universal gravitation:

$F_g = G\frac{m_1 m_2}{r^2}$

where $G = 6.674 \times 10^{-11} \, \text{N·m}^2/\text{kg}^2$.

Near Earth's surface:

$F_g = mg$

where $g = 9.8 \, \text{m/s}^2$.

Normal Force

The normal force $F_N$ acts perpendicular to a surface. For an object on a flat surface:

$F_N = mg$

Friction

Static friction:

$f_s \leq \mu_s F_N$

Kinetic friction:

$f_k = \mu_k F_N$

where $\mu_s$ and $\mu_k$ are the coefficients of static and kinetic friction.

Spring Force (Hooke's Law)

$F_s = -kx$

where $k$ is the spring constant and $x$ is the displacement from equilibrium.

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Work and Energy

Work

Work done by a constant force:

$W = Fd\cos\theta$

where $\theta$ is the angle between the force and displacement vectors.

Kinetic Energy

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

Potential Energy

Gravitational potential energy:

$PE_g = mgh$

Elastic potential energy:

$PE_s = \frac{1}{2}kx^2$

Work-Energy Theorem

$W_{net} = \Delta KE = KE_f - KE_i$

Conservation of Energy

$KE_i + PE_i = KE_f + PE_f$

$\frac{1}{2}mv_i^2 + mgh_i = \frac{1}{2}mv_f^2 + mgh_f$

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Momentum and Impulse

Linear Momentum

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

Impulse

$\vec{J} = \vec{F}\Delta t = \Delta\vec{p}$

Conservation of Momentum

For an isolated system:

$m_1\vec{v}_{1i} + m_2\vec{v}_{2i} = m_1\vec{v}_{1f} + m_2\vec{v}_{2f}$

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Interactive Simulation: Forces and Motion

Experiment with the simulation below to see Newton's laws in action. Apply different forces to objects and observe how they accelerate.

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3D Molecular Viewer

Explore the 3D structure of Crambin (PDB: 1CRN), a small plant protein with 46 amino acids. Drag to rotate, scroll to zoom, and right-click to pan.

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Practice Questions

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Interactive Exercises

Drag-and-Drop: Order the Kinematic Steps

Put these steps in the correct order for solving a kinematics problem:

Matching: Newton's Laws

Match each law of motion with its correct statement:

Fill in the Blanks: Energy Conservation

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• Kinematics equations describe motion with constant acceleration: $v = v_0 + at$ and $x = x_0 + v_0 t + \frac{1}{2}at^2$
• Newton's Second Law: $\sum \vec{F} = m\vec{a}$
• Work: $W = Fd\cos\theta$; Kinetic Energy: $KE = \frac{1}{2}mv^2$
• Energy is conserved: $KE_i + PE_i = KE_f + PE_f$
• Momentum is conserved in isolated systems