UNIVERSIDAD LIBREFACULTAD DE INGENIEIRA

DEPARTAMENTO DE CIENCIAS BASICAS

LECTURAS EN INGLES

FISICA MECANICA

Introduction to Vectors

In order to represent physical quantities such as position and momentum in

more than one dimension, we must introduce new mathematical objects called

vectors. Technically speaking, a vector is defined as an element of a vector

space, but since we will only be dealing with very special types of vector spaces

(namely, two- and three-dimensional Euclidean space) we can be more specific.

For our purposes, a vector is either an ordered pair or triplet of numbers. On a

two-dimensional plane, for instance, any point (a, b) is a vector. Graphically, we

often represent such a vector by drawing an arrow from the origin to the point,

with the tip of the arrow resting at the point. The situation for three-dimensional

vectors is very much the same, with an ordered triplet (a, b, c) being

represented by an arrow from the origin to the corresponding point in three-

dimensional space.

The vector (a, b) in the Euclidean plane.

Unlike scalars, which have only a value for magnitude, vectors are often

described as objects that have both magnitude and direction. This can be seen

intuitively from the arrow-like representation of a vector in the plane. The

magnitude of the vector is simply the length of the arrow (i.e. the distance from

the point to the origin), and can be easily computed using the Pythagorean

Theorem. The direction of a vector in two dimensions can be characterized by a

single angle θ; the direction of a vector in three dimensions can be specified

using two angles (usually denoted θ and φ).

1. Search since the theorem of Pythagoras is applied for the measurement of

the magnitude of a vector (a.b).

2. Represent in a three-dimensional plane the vector that goes from the origin to

the point (3,4,6) and calculate the angles that it forms using the trigonometrical

relations.

INFOGRAFIA

www.sparknotes.com/physics

UNIVERSIDAD LIBREFACULTAD DE INGENIEIRA

DEPARTAMENTO DE CIENCIAS BASICAS

LECTURAS EN INGLES

FISICA MECANICA

Position Functions in One DimensionIn order to describe the motion of an object we must be to determine the

position of the object at any point in time. In other words, if we are given the

problem of describing the motion of an object, we will have reached a solution

when we find a position function, x(t), which tells us the position of that object at

any moment in time. (Note that "t" is usually understood to be a time variable,

so in writing the position function "x" as "x(t)" we are explicitly indicating that

position is a function of time.) There are a variety of functions that can

correspond to the position of moving objects. In this section we will introduce

some of the more common ones that tend to arise in basic physics problems.

Examples of Position Functions

1. ctx =)( , where c is a constant. As you might expect, an object that has this

as its position function isn't going anywhere. At all times its position is exactly

the same: c.

2. cVttx +=)( , where v and c are constants. An object with this position

function starts off (at t = 0) with a position c, but its position changes with time.

At a later time, say t = 5, the object's new position will be given by cVx += 5)5( .

Because the exponent of t in the above equation is 1, we say the object

changes linearly with time. Such objects are moving at a constant velocity

(which is why the coefficient of "t" has been suggestively labeled V).

3. 2

21)( attx = , where a is a constant. At t = 0, this object is situated at the

origin, but its position changes quadratically with time (since the exponent of t in

the above equation is 2). For positive a, the graph of this position function looks

like a parabola that touches the horizontal axis (the time-axis) at the point t = 0.

For negative values of a, the graph of this function is an upside-down parabola.

Such a position function corresponds to objects undergoing constant

acceleration (which is why the coefficient of " 2t " has been conveniently written

as a21

).

4. Coswttx =)( , where w is a constant. An object with this position function is

undergoing simple harmonic motion, which means its position is oscillating back

and forth in a special fashion. Since the range of the cosine function is (- 1, 1),

the object is constrained to move within this small interval and will forever be

retracing its path. An example of such an object is a ball hanging from a spring

that is bouncing up and down. In contrast to the above three examples, this kind

of function describes motion where neither the position, velocity, nor

acceleration of the object are constant.

EXERCISE

Make the graphic for each example, using values between 0 at 10 meters.

INFOGRAFIA

www.sparknotes.com/physics

UNIVERSIDAD LIBREFACULTAD DE INGENIEIRA

DEPARTAMENTO DE CIENCIAS BASICAS

LECTURAS EN INGLES

FISICA MECANICA

Velocity, in One DimensionIn one dimension, velocity is almost exactly the same as what we normally call

speed. The speed of an object (relative to some fixed reference frame) is a

measure of "how fast" the object is going and coincides precisely with the idea

of speed that we normally use in reference to a moving vehicle. Velocity in one

dimension takes into account one additional piece of information that speed,

however, does not: the direction of the moving object. Once a coordinate axis

has been chosen for a particular problem, the velocity V of an object moving at

a speed s will either be V = s, if the object is moving in the positive direction, or

V = - s, if the object is moving in the opposite (negative) direction. More

explicitly, the velocity of an object is its change in position per unit time, and is

hence usually given in units such as sm

(meters per second) or hKm

(kilometers

per hour). The velocity function, )(tV , of an object will give the object's velocity

at each instant in time just as the speedometer of a car allows the driver to see

how fast he is going. The value of the function V at a particular time 0t is also

known as the instantaneous velocity of the object at time 0tt = , although the

word "instantaneous" here is a bit redundant and is usually used only to

emphasize the distinction between the velocity of an object at a particular

instant and its "average velocity" over a longer time interval. (Those familiar with

elementary calculus will recognize the velocity function as the time derivative of

the position function.)

Average Velocity

Now that we have a better grasp of what velocity is, we can more precisely

define its relationship to position. We begin by writing down the formula for

average velocity. The average velocity of an object with position function )(tx

over the time interval ),( 10 tt is given by:

01

01

ttXX

Vavg −−

=

In other words, the average velocity is the total displacement divided by the total

time. Notice that if a car leaves its garage in the morning, drives all around town

throughout the day, and ends up right back in the same garage at night, its

displacement is 0, meaning its average velocity for the whole day is also 0.

1. Calculate the average speed in the tour from your house up to the university

and describe since it obtained the magnitudes of distance and time.

2. Describe the process to turn the units of sm

to hKm

INFOGRAFIA

www.sparknotes.com/physics

UNIVERSIDAD LIBREFACULTAD DE INGENIEIRA

DEPARTAMENTO DE CIENCIAS BASICAS

LECTURAS EN INGLES

FISICA MECANICA

Acceleration in One Dimension

Just as velocity is given by the change in position per unit time, acceleration is

defined as the change in velocity per unit time, and is hence usually given in

units such as 2sm

Free Fall

In general, the acceleration of an object in the earth's gravitational field is not

constant. If the object is far away, it will experience a weaker gravitational force

than if it is close by. Near the surface of the earth, however, the acceleration

due to gravity is approximately constant and is the same value regardless of the

mass of the object (i.e, in the absence of friction from wind resistance, a feather

and a grand piano fall at exactly the same rate). This is why we can use our

equations for constant acceleration to describe objects in free fall near the

earth's surface. The value of this acceleration is 28,9sma = . From now on,

however, we will denote this value by g, where g is understood to be the

constant (Notice that this is not valid at large distances from the surface of the

earth: the moon, for instance, does not accelerate towards us at 28.9sm

.). The

equations describing an object moving perpendicular to the surface of the earth

(i.e. up and down) are now easy to write. If we locate the origin of our

coordinates right at the earth's surface, and denote the positive direction as that

which points upwards, we find that:

002

21 xtVgtx ++−=

Notice the - sign that arises because the acceleration due to gravity points

downwards, while the positive position direction was chosen to be up. How

does this relate to an object in free fall? Well, if you stand at the top of a tower

with height h and let go of an object, the initial velocity of the object is 00 =V ,

while the initial position is hx =0 . Plugging these values into the above equation

we find that the motion of an object falling freely from a height h is given by:

hgtx +−= 2

21

If we want to know, for instance, how long it takes for the object to reach the

ground, we simply set 0=x and solve for t. We find that at t, the object hits

the ground (i.e. reaches the position 0).

1. There defines each of the variables that intervene in both mentioned

equations

2. Mention three essential characteristics of the movement of free fall.

INFOGRAFIA

www.sparknotes.com/physics

UNIVERSIDAD LIBREFACULTAD DE INGENIEIRA

DEPARTAMENTO DE CIENCIAS BASICAS

LECTURAS EN INGLES

FISICA MECANICA

WORKWork, though easily defined mathematically, takes some explanation to grasp

conceptually. In order to build an understanding of the concept, we begin with

the most simple situation, then generalize to come up with the common formula.

The Simple Case

Consider a particle moving in a straight line that is acted on by a constant force

in the same direction as the motion of the particle. In this very simple case, the

work is defined as the product of the force and the displacement of the particle.

Unlike a situation in which you hold something in place, exerting a normal force,

the crucial aspect to the concept of work is that it defines a constant force

applied over a distance. If a force F acts on a particle over a distance x, then

the work done is simply:

FxW =

Since W increases as x increases, given a constant force, the greater the

distance during which that force acts on the particle, the more work is done. We

can also see from this equation that work is a scalar quantity, rather than a

vector one. Work is the product of the magnitudes of the force and the

displacement, and direction is not taken into account.

The joule is a multipurpose unit. It serves not only as a unit of work, but also of

energy. Also, the joule is used beyond the realm of physics, in chemistry, or any

other subject dealing with energy. If a force is to do work, it must act on a

particle while it moves; it cannot just cause it to move. For instance, when you

kick a soccer ball, you do no work on the ball. Though you produce a great deal

of motion, you have only instantaneous contact with the ball, and can do no

work. On the other hand, if I pick the ball up and run with it, I do work on the

ball: I am exerting a force over a certain distance. In technical jargon, the "point

of application" of the force must move in order to do work.

1. Why does not normal force to the direction of the movement to do work?.

2. Which is the work realized to move a body of 1 Kg of mass, along 1 m?

3. How is Joule defined?

INFOGRAFIA

www.sparknotes.com/physics

UNIVERSIDAD LIBREFACULTAD DE INGENIEIRA

DEPARTAMENTO DE CIENCIAS BASICAS

LECTURAS EN INGLES

FISICA MECANICA

READING FOR MECHANICAL PHYSICS

Galileo's measurements were used as base for Newton to obtain his

movement laws. In Galileo's experiments, when a body was falling down

rolling, It acted its same force (weight) and the effect that was taking place,

consisted of constant acceleration. This was demonstrating that the real

effect of a force, was changing the speed of the body, instead of simply

putting it in movement, as it was thought previously. It also was meaning

that if there was no force exerting on a body, this one would remain moving

in a straight line at constant speed. This idea was formulated explicitly for

the first time by the Newton's Mathematics principles, published in 1687, and

it is known as The Newton's first law. This one affirms that the body will

accelerate or change its speed at a proportional rate to the exerted force.

(For example the acceleration will double when the applied force is the

double). At the same time, the acceleration will diminish when the body’s

mass increase. (The same force acting on a double mass body, in

proportion to a second one, will produce a half of the acceleration of the

second one.

QUESTIONS:

1. Do forces make magnitude of the speed to change?

2. How does acceleration of a body varies when a force acts on it?