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 | = a \cos \beta\left | \vec{d} \right | = \vec{a} \cdot \vec{d} = \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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Forest_Ugrad_ClassicalMechanics