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] c[/math]) nor are microscopically small ( [math]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]\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]\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]\frac{\vec{a} \cdot \vec{b}}{\left | \vec{b} \right |}[/math] is the length of [math]\vec{a}[/math] that is along the direction of [math]\vec{b}[/math] (a projection like the casting of a shadow)

Commutative property of scalar product

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

proof

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

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

Distributive property of scalar product

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

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

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

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

physical interpretation

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

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

then the area of the parallelogram is given by

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

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

thus

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

proof

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

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

Distributive property of the vector product

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

physical interpretation

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

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

if

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

then

[math]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]h=a \cos \beta[/math]

[math]V= h \left | \vec{d} \right |[/math] = a \cos \beta\left | \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.

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