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.
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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.
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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
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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.
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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?
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LECTURAS EN INGLES
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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?