Forest UCM NLM

Newton's Laws of Motion

Limits of Classical Mechanic

Classical Mechanics is the formulations of physics developed by Newton (1642-1727), Lagrange(1736-1813), and Hamilton(1805-1865).

It may be used to describe the motion of objects which are not moving at high speeds (0.1[math]\displaystyle{ c }[/math]) nor are microscopically small ( [math]\displaystyle{ 10^{-9} m }[/math]).

The laws are formulated in terms of space, time, mass, and force:

Vectors

Vector Notation

A vector is a mathematical construct of ordered elements that represent magnitude and direction simultaneously.

[math]\displaystyle{ \vec{r} = x \hat{i} + y \hat{j} + z \hat{k} = (x,y,z) = \sum_1^3 r_i \hat{e}_i }[/math]

Vectors satisfy the commutative (order of addition doesn't matter) and associative ( doesn't matter which you add first) properties.

The multiplication of two vectors is not uniquely defined. At least three types of vector products may be defined.

Scalar ( Dot ) product

definition

[math]\displaystyle{ \vec{a} \cdot \vec{b} = \left | a \right | \left | b \right | cos \theta = a_1 b_1 + a_2 b_2 + a_3 b_3 }[/math]

physical intepretation

[math]\displaystyle{ \frac{\vec{a} \cdot \vec{b}}{\left | \vec{b} \right |} }[/math] is the length of [math]\displaystyle{ \vec{a} }[/math] that is along the direction of [math]\displaystyle{ \vec{b} }[/math] (a projection like the casting of a shadow)

Commutative property of scalar product

[math]\displaystyle{ \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} }[/math]

proof

[math]\displaystyle{ \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 }[/math]	definition of dot product
[math]\displaystyle{ a_1 b_1 + a_2 b_2 + a_3 b_3=b_1 a_1 + b_2 a_2 + b_3 a_3 }[/math]	comutative property of multiplication
[math]\displaystyle{ b_1 a_1 + b_2 a_2 + b_3 a_3=\vec{b} \cdot \vec{a} }[/math]	definition of dot product

[math]\displaystyle{ \vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{a} }[/math]

Distributive property of scalar product

[math]\displaystyle{ \vec{a} \cdot \left ( \vec{b} + \vec{c} \right ) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} }[/math]

Vector ( Cross ) product

definition

[math]\displaystyle{ \vec{a} \times \vec{b} = \left( a_2b_3-a_3b_2\right) \hat{e}_1 +\left( a_3b_1-a_1b_3\right) \hat{e}_2 +\left( a_1b_2-a_2b_1\right) \hat{e}_3 }[/math]

The vector product of [math]\displaystyle{ \vec{a} }[/math] and [math]\displaystyle{ \vec{b} }[/math] is a third vector [math]\displaystyle{ \vec{c} }[/math] with the following properties.

[math]\displaystyle{ \left | \vec{c} \right | = \left | \vec{a} \right | \left | \vec{b} \right | \sin \theta }[/math]

[math]\displaystyle{ \vec{c} }[/math] is [math]\displaystyle{ \perp }[/math] to [math]\displaystyle{ \vec{a} }[/math] and [math]\displaystyle{ \vec{b} }[/math]

the right hand rule convention is used to determine the direction of [math]\displaystyle{ \vec{c} }[/math]

physical interpretation

[math]\displaystyle{ A = \left | \vec{a} \times \vec{b} \right | = }[/math] area of a parallelogram with vectors [math]\displaystyle{ \vec{a} }[/math] and [math]\displaystyle{ \vec{b} }[/math] forming adjacent edges

let [math]\displaystyle{ h }[/math] represent the perpendicular distance from the teminus of [math]\displaystyle{ \vec{b} }[/math] to the line of action of [math]\displaystyle{ \vec{a} }[/math] ( a.k.a. the height)

then the area of the parallelogram is given by

[math]\displaystyle{ A=\left | \vec{a} \right | h }[/math]

the height [math]\displaystyle{ h }[/math] is equivalent to [math]\displaystyle{ \left | \vec{b} \right | \sin \theta }[/math] where [math]\displaystyle{ \theta }[/math] is the angle between the vectors [math]\displaystyle{ \vec{a} }[/math] and [math]\displaystyle{ \vec{b} }[/math]

thus

[math]\displaystyle{ A=\left | \vec{a} \right | h = \left | \vec{a} \right | \left ( \left | \vec{b} \right | \sin \theta \right ) = \left | \vec{a} \times \vec{b} \right | }[/math]

NON-Commutative property of vector product

[math]\displaystyle{ \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} }[/math]

proof

[math]\displaystyle{ \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 }[/math]	definition of dot product
[math]\displaystyle{ a_1 b_1 + a_2 b_2 + a_3 b_3=b_1 a_1 + b_2 a_2 + b_3 a_3 }[/math]	comutative property of multiplication
[math]\displaystyle{ b_1 a_1 + b_2 a_2 + b_3 a_3=\vec{b} \cdot \vec{a} }[/math]	definition of dot product

[math]\displaystyle{ \vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{a} }[/math]

Distributive property of the vector product

[math]\displaystyle{ \vec{a} \times \left ( \vec{b} + \vec{c} \right ) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} }[/math]

The scalar triple product

definition

scalar triple product [math]\displaystyle{ \equiv \vec{a} \cdot \left (\vec{b} \times \vec{c} \right ) }[/math]

physical interpretation

the volume of a parallelpiped with the vectors [math]\displaystyle{ \vec{a} }[/math], [math]\displaystyle{ \vec{b} }[/math], [math]\displaystyle{ \vec{c} }[/math] forming adjacent edges is given by

[math]\displaystyle{ V = \left | \vec{a} \cdot \left (\vec{b} \times \vec{c} \right ) \right | }[/math]

if

[math]\displaystyle{ \vec{d} \equiv \vec{b} \times \vec{c} = }[/math] Area vector of the parallelpiped base

then

[math]\displaystyle{ V = h \left | \vec{d} \right | }[/math]

as shown in a description of the dot product, the height of the parallelpiped can be written as

[math]\displaystyle{ h=a \cos \beta }[/math]

[math]\displaystyle{ V= h \left | \vec{d} \right | = a \cos \beta\left | \vec{d} \right | = \left | \vec{a} \cdot \vec{d}\right | = \left | \vec{a} \cdot \left (\vec{b} \times \vec{c} \right ) \right | }[/math]

A third vector product is the tensor direct product.

Space and Time

Space

Cartesian, Spherical, and Cylindrical coordinate systems are commonly used to describe three-dimensional space.

Forest_UCM_NLM_Ch1_CoordSys

Time

In classical mechanic, unlike relativistic mechanics, all observers agree on the times of all event.

Reference frames

A description of systems that obey classical mechanics will involve making a choice of a frame of reference from which the system will be described.

In most cases you will prefer to use a non-accelerating (inertial) reference system oriented to simplify the description of the object that is in motion. Newton's laws of motion are obeyed in a reference frame that is accelerating or rotating.

Newton's Laws

1st law

Newton's Principia pg 83

"Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon."

Taylor's Classical Mechanics

"In the absence of forces, a particle moves with constant velocity [math]\displaystyle{ v }[/math]."

2nd Law

Newton's Principia pg 83

"The alteration of motion is ever proportional to the motive force impressed ; and is made in the direction of the right line in which that force is impressed."

Taylor's Classical Mechanics

"For any particle of mass [math]\displaystyle{ m }[/math], the net force [math]\displaystyle{ F }[/math] on the particle is always equal to the mass [math]\displaystyle{ m }[/math] times the particle's acceleration."

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

3rd Law

Newton's Principia pg 83

"To every action there is always opposed an equal reaction : or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. "

Taylor's Classical Mechanics

If object 1 exerts a force [math]\displaystyle{ \vec{F}_{21} }[/math] on object 2, then object 2 always exerts a reaction force [math]\displaystyle{ \vec{F}_{12} }[/math] on object 1 given by

[math]\displaystyle{ \vec{F}_{12} -\vec{F}_{21} }[/math]

Forest_Ugrad_ClassicalMechanics