# 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) nor are microscopically small ( ).

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.

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

physical intepretation
is the length of that is along the direction of (a projection like the casting of a shadow)

### Commutative property of scalar product

proof
 definition of dot product comutative property of multiplication definition of dot product

## Vector ( Cross ) product

definition

The vector product of and is a third vector with the following properties.

is to and
the right hand rule convention is used to determine the direction of
physical interpretation
area of a parallelogram with vectors and forming adjacent edges

let represent the perpendicular distance from the teminus of to the line of action of ( a.k.a. the height)

then the area of the parallelogram is given by

the height is equivalent to where is the angle between the vectors and

thus

### The scalar triple product

definition
scalar triple product
physical interpretation
the volume of a parallelpiped with the vectors , , forming adjacent edges is given by

if

Area vector of the parallelpiped base

then

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

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.

## 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.

Galilean Transformations

# 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 ."

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 , the net force on the particle is always equal to the mass times the particle's acceleration."
More explicit version
The vector sum of the external forces acting on the system is equl to its mass times its acceleration.

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

the dot over 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

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.

to simply the notation we let

Then

integrating one more time with the same assumption

## 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 on object 2, then object 2 always exerts a reaction force on object 1 given by

### Conservation of Momentum

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

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.