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 | = 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 (1687 published in latin, translated to english in 1726) 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]v[/math]."

Examples

1. You hit the brakes and lock up the wheels of your car when you hit a patch of ice during your winter drive. You stay in motion.
2. The coffee in your coffee cup stays at the brim while driving in your car until you hit the brakes and spill cofee all over the floor boards.Coffe that isn't held back by the brim of the cup stays in motion.
3. A skateboarder's skateboard hits the curb while he is riding it and he flys forward off the board.
4. If there is no friction between the table cloth and the dishes, you can just pull on the table cloth and the dishes will stay in place (but the table may move if there is friction between the table cloth and the table).

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]m[/math], the net force [math]F[/math] on the particle is always equal to the mass [math]m[/math] times the particle's acceleration."

[math]\vec{F} = m \vec{a}[/math]

More explicit version

The vector sum of the external forces acting on the system is equl to its mass times its acceleration.

[math]\sum \vec{F}_{ext} = m \vec{a}[/math]

Newton's second law may be expressed as a first order differential equation such that

[math]\sum \vec{F}_{ext} = \frac{d\vec{p}}{dt} = \vec \dot{p}[/math]

the dot over [math]\vec{p}[/math] is short hand notation for taking a derivative with respect to time.

The above form is particularly useful for problems that involve mass that changes as a funtion of time (rocket problems)

Newton's second law is a second order differential equation given by

[math]\vec{a} = \frac{d^2 \vec{x}}{dt^2} = \vec \ddot{x} =\frac{\sum \vec{F}_{ext}}{m}[/math]

The above 2nd order differential equation can be readily solved if the sum of the external forces and the mass do not depend on time or if their ratio is a constant in time.

[math]\vec v = \int_{t_i}^{t_f} \vec \dot v dt = \int_{t_i}^{t_f} \frac{\sum \vec{F}_{ext}}{m} dt[/math]

[math] \vec v(t_f) - \vec v(t_i) = \frac{\sum \vec{F}_{ext}}{m} \left (t_i-t_f\right)[/math]

[math] \vec v(t_f) = \vec v(t_i) + \frac{\sum \vec{F}_{ext}}{m} \left (t_i-t_f\right)[/math]

to simply the notation we let

[math]t_i \equiv 0 \;\; t_f \equiv f [/math]

[math]\vec{v}(t_i) \equiv \vec{v}_0[/math]

Then

[math] \vec v(t) = \vec{v}_0 + \frac{\sum \vec{F}_{ext}}{m} t[/math]

integrating one more time with the same assumption

[math]\vec{r}(t) = \int_0^t \vec{r} dt = \int_0^t \left ( \vec{v}_0 + \frac{\sum \vec{F}_{ext}}{m} t \right )[/math]

[math]\vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{\sum \vec{F}_{ext}}{2m} t^2[/math]

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

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

Conservation of Momentum

Conservation of momentum is an interpretation of Newton's third law

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

[math]\vec{F}_{12} +\vec{F}_{21} = 0[/math]

[math]\frac{d\vec{p}_{12}}{dt} +\frac{d\vec{p}_{21}}{dt} = 0[/math]

[math]\frac{d}{dt} \left ( \vec{p}_{12} +\vec{p}_{21}\right ) = 0[/math]

[math] \left ( \vec{p}_{12} +\vec{p}_{21}\right ) = Constant[/math]

The total momentum of a system of a two particle with no external force is conserved.

The above may be generalized to any system of many particles which has no NET EXTERNAL force.

Sample Problems

Atwoods Machine

Forest_UCM_NLM_AtwoodMachine

Block on incline with friction

Forest_UCM_NLM_BlockOnInclineWfriction

Forest_UCM_NLM_BlockOnIncline

Oscillatiions

Forest_UCM_NLM_Oscilations

Galilean Transformations

Forest_UCM_NLM_GalileanTans

Forest_Ugrad_ClassicalMechanics