Preface

Prerequisites

Learning ethics

Introduction

What is 🍎Classic mechanics ?

Why does 🍎Classic mechanics matter to you?

Ecosystem

Standards, jobs, industry, roles, and research

Story

FAQ

Worked examples

Model framework

Subsection

Subsubsection

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

Vectors

Subsection

Subsubsection

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

One dimensional kinematics

Subsection

Subsubsection

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

Two dimensional kinematics

Vector Description of Motion in Two Dimension

In this section we introduce position, velocity, and acceleration that moves in $k$-dimensions by treating each vector component independently. In Cartesian coordinates, the position vector $\textbf{r}(t)$ with respect to some choice for the object at time $t$ is given by a parametrized vector as follows:

The velocity vector $\textbf{v}(t)$ at time $t$ is the derivative of the position vector,

Similarly, the acceleration vector $a(t)$ is defined in a similar fashion as derivative of the velocity vector,

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

Circular motion

Subsection

Subsubsection

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

Newton’s law of motion

Subsection

Subsubsection

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

Applications of Newton’s Second Law

Subsection

Subsubsection

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

Circular Motion Dynamics

Subsection

Subsubsection

Exercises and Projects

Engineering

Crankshaft mechanism

Reciprocating motion

Crank (mechanism)

Hierapolis sawmill

Summary

Key decisions

FAQ

Reference Notes

Momentum, System of Particles, and Conversation of Momentum

System

$state_1\to state_2$ such that the momentum remains constant.

$property(t)= property(t+1)$

$p(t)=mv(t)$

$p(t)=p(t+1)$

The second Newton’s law states that $F=\dfrac{d}{dt} p=ma$

$p_1(t)+p_2(t)+...=p_1(t+1)+p_2(t+1)+...$

$m_1v_1(t)+m_2v_2(t)=m_1v_1(t+1)+m_2v_2(t+1)$

$m_1v_{1,A}+m_2v_{2,A}=m_1v_{1,B}+m_2v_{1,B}$

$m_1\textbf{v(t)}_1+m_2\textbf{v(t)}_2+...=m_1\textbf{v(t+1)}_1+m_2\textbf{v(t+1)}_2+...$

Let $V$ a matrix of velocities and a vector of masses as follows

$V(t)=\begin{pmatrix} \textbf{ v(t)}_1\\ \textbf{ v(t)}_2\\ \textbf{ v(t)}_3\\ ... \end{pmatrix}$ and

$\textbf{m}=\begin{bmatrix} m_1,m_2,m_3,... \end{bmatrix}$

$V^T\textbf{m}=m_1\textbf{v}_1+m_2\textbf{v}_2+...$

$V^T(t)\textbf{m}=V^T(t+1)\textbf{m}$

$(V^T(t)-V^T(t+1))\textbf{m}=0$

subject to the kind of collision.

For instance, if the collision is elastic, the kinetic energy is constant.

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

Reference Frames

Subsection

Subsubsection

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

Momentum and the Flow of Mass

Subsection

Subsubsection

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

Energy, Kinetic, and Work

Subsection

Subsubsection

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

Potential Energy and Conversation of Energy

Subsection

Subsubsection

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

Collision Theory

Subsection

Subsubsection

Exercises and Projects

Summary

Key decisions

FAQ

Reference Notes

Tutorials

Next steps

Units

Billon is $10^9$ in English, Portuguese, and Greek -"Mil millones" in Spanish. But, It's $10^{12}$ in Spanish -trillion in English.

References

TODO

Motion

1. $\quad v=v_0+at \quad$, falta el desplazamiento ($\Delta x$)

2. $\quad {\Delta x}=(\dfrac{v+v_0}{2})t$, falta la aceleracion $(a)$

3. $Δx=v_0​t+\dfrac{1}{2}at^2$, falta v

4. $\quad v^2=v_0^2+2a$, falta t

Kinetic and Static Friction

Example 1.

Circular Motion Dynamics

Resistive forces

When a solid object moves through a fluid (liquid or gas) it will experience a resistive force, called the drag force, opposing its motion. This force depends on both the properties of the object and the properties of the fluid. It also depends on the the density, viscosity, and compressibility of the fluid.

Resistive forces - low speed case

Linear momentum

Linear momentum is a vector quantity, product of an object's mass and velocity. Also called “momentum” for short. Momentum describes the amount of mass in motion.

How momentum and net force are related

Forces cause a change in momentum, but momentum does not cause a force. The bigger the change in momentum, the more force you need to apply to get that change in momentum.

Impulse

Impulse (symbolized by J or Imp) is the integral of a force, F, over the time interval, t, for which it acts. Since force is a vector quantity, impulse is also a vector quantity. Impulse applied to an object produces an equivalent vector change in its linear momentum, also in the same direction.

Angular force

Torque

Torque, moment, moment of force, rotational force, or "turning effect" (τ) is a measure of the force that can cause an object to rotate about an axis -Newtons*meters, no joule. An object at rest remains at rest (not rotating); an object rotating, continues to rotate with constant angular velocity; unless acted on by an external torque.

One can calculate the torque exerted by a force around an axis of rotation via

where
    torque is the torque, in Newton-meters
    force  is the force applied, in Newtons
    a lever arm is the "effective distance" from the point 
    of force to the axis of rotation

Just as a force is what causes an object to accelerate in linear kinematics, torque is what causes an object to acquire angular acceleration.

Torque is a vector quantity. The direction of the torque vector depends on the direction of the force on the axis.

Anyone who has ever opened a door has an intuitive understanding of torque. When a person opens a door, they push on the side of the door farthest from the hinges. Pushing on the side closest to the hinges requires considerably more force. Although the work done is the same in both cases (the larger force would be applied over a smaller distance) people generally prefer to apply less force, hence the usual location of the door handle.

Rotational Equilibrium

Worked examples

1. Una clavadista de 582 N de peso esta en la punta de un trampolín uniforme de 4.48m y de 142 N. El trampolín esta sostenido por dos pedestales separados una distancia de 1.55m. Calcule la fuerza de compresión en los dos pedestales.

Planning

1. We draw diagrams.

First, the effective distance is L=4.48m. Thus, La=0, La=1.55m...

1. We'll use Newton's laws:

   $$\sum{F}=c$$

2. Solve for Fb.

   $$\sum{\tau}=0 \text{ by rotational equlibrium}$$

   $$\tau_a+\tau_b+\tau_p+\tau_w=0$$

   $$(F_aL_a)+(F_bL_b)-(W_pL_p)-(WL)=0, \text{ by } \theta=pi/2 \text{ and this is bi-plane}$$

   $$0+(1.55m\times F_b)-(2.24m\times 142N)-(4.48m\times 582N)=0$$

   $$F_b=1887.38N$$

   3. Solve for Fa

   $$\sum{F_y}=0 \text{ by translational equilibrium}$$

   $$F_a+F_b-W_p-W=0$$

   $$F_a+1887.38N-142N-582N=0$$

   $$F_a=-1163.38N$$

   2. Two identical uniform frictionless spheres, each of weight W. Rest at the bottom of a fixed container. The line of centers of the spheres makes an angle θ with the horizontal. What is the magnitude of the force exerted on the spheres by the container bottom? What is the magnitude of the force exerted on the spheres by one side of the container? What is the magnitude of the forces exerted on the spheres by one another?

   

   1. We draw free-body diagrams.

   

   1. Modelling each shephere.

      Upper sphere.

      $$\sum F_x =N_2cos(\theta)-N_1=0(I)$$

      $$\sum F_y = N_{2}sin(\theta)-W=0(II)$$

   Lower shpere.

   $$\sum F_x=N_1-N_2cos(\theta)=0(III)$$

   $$\sum F_y = N_3-W-N_2sin(\theta)=0(IV)$$

   1. Solve for $N_3$ in terms W (Spheres to bottom container).

      $$N_2sin(\theta)=W (II)$$

      $$N_3=W+N_2sin(\theta)=W+W(IV)$$

      $$N_3=2W$$

   2. Solve for $N_1$ in terms W (Shperes to one side of the container).

      $$N_2=\dfrac{N_1}{cos(\theta)}(I)$$

      $$N_2=\dfrac{W}{sin(\theta)}(II)$$

      $$\dfrac{N_1}{cos(\theta)}=\dfrac{W}{sin(\theta)}$$

      $$N_1=\dfrac{cos(\theta)W}{sin(\theta)}$$

      $$N_1=cot(\theta)W$$

   3. Solve for $N_2$ in terms W (Spheres by one another).

      $$N_2=\dfrac{W}{sin(\theta)}(II)$$

      $$N_2=csc(\theta)W$$