Download Physics for Engineers and Scientists Spring 2017

Physics for Engineers and Scientists
Spring 2017
Mario C Dı́az
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1.1
Lesson 6
Newton’s laws
Why do objects move?
The ancient Greeks were the first to think about the nature of motion. What is that set things in
motion?
Aristotle in his book Physics said:
“Of things that exist, some exist by nature, some from other causes. By nature the animals and
their parts exist, and the plants and the simple bodies (earth, fire, air, water)-for we say that these
and the like exist by nature. All the things mentioned present a feature in which they differ from
things which are not constituted by nature. Each of them has within itself a principle of motion and
of stationariness (in respect of place, or of growth and decrease, or by way of alteration). On the
other hand, a bed and a coat and anything else of that sort, qua receiving these designations i.e. in
so far as they are products of art-have no innate impulse to change. But in so far as they happen to
be composed of stone or of earth or of a mixture of the two, they do have such an impulse, and just
to that extent which seems to indicate that nature is a source or cause of being moved and of being
at rest in that to which it belongs primarily, in virtue of itself and not in virtue of a concomitant
attribute”1 And he adds:
“Nature then is what has been stated. Things have a naturewhich have a principle of this kind. Each
of them is a substance; for it is a subject, and nature always implies a subject in which it inheres.
The term according to nature is applied to all these things and also to the attributes which belong
to them in virtue of what they are, for instance the property of fire to be carried upwards-which is
not a nature nor has a nature but is by nature or according to nature.” 2
What Aristotle is claiming is that 1) there is a natural order of things. Heavens are eternal
and immobile. It is made according to many Greek philosophers before him, a theory which he
accepted, of a substance completely different to the nature of things on Earth. Things on Earth
1
Aristotle (2007-03-02). Aristotle’s Collection [ 29 Books] (Kindle Locations 26989-26997). Publish This, LLC.
Kindle Edition.
2
Aristotle (2007-03-02). Aristotle’s Collection [ 29 Books] (Kindle Locations 27002-27006). Publish This, LLC.
Kindle Edition.
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were made of a combination of four elements: earth, fire, air and water. When taking objects out
of rest they will tend to move according to their composition. For Aristotle a heavier object, would
reach the ground faster than a lighter one when dropped from the same level above the ground at
the same time. He also claimed that a force must be constantly applied to keep something moving.
Both of these concepts were wrong, but it would take almost two thousand years to understand it!
The first step in the critic to Aristotle’s ideas came in the 16th century when Nicolaus Copernicus published his sun-centered model of the universe. Aristotle theorized that the sun, the moon
and the planets all revolved around Earth on a set of celestial spheres. Copernicus proposed that
the planets of the solar system revolved around the sun, not Earth. The immutabilty of Heavens
was precisely based on the earth being at the center of the Universe.
Soon after Copernicus Galileo was the first one in making experiments (some practically, some
as thought-experiments) that let him deduct that the acceleration of gravity was the same for all
bodies in the absence of air-friction (see video clip of moon experiment). Galileo was the first
scientist to define acceleration as the time rate change of velocity: a crucial concept. On his first
attempts he defined it as the change of velocity with distance. Such a definition would lead to the
trajectories being exponentials, not parabolas.
a=
dv
=g
dx
where g = −9.8m/s2 From there you get
Z
Z
dv = gdx → dv = gdx → v = gx + const.
(1)
(2)
But this is
dx
= gx + const.
dt
(3)
And from there we get:
dx
= gdt →
x
Z
dx
=
x
Z
gdt → ln x = gt
(4)
where ln is the natural logarithm. We can take the exponential of both sides to get:
x(t) = exp(gt) = e−9.8t
(5)
And of course these curves are very different from parabolas. It is interesting to plot the two
solutions that could be obtained with a simple case. Let’s look at a stone drop vertically from a 10
meters height. The two solutions are:
x(t) = 10 exp(gt) = 10e−9.8t
2
(6)
Figure 1: An object dropped from 10m above the ground falling exponentially -blue- and parabolically -redand
9.8 2
t
(7)
2
The exponential curve in Figure 1, does not correspond to the actual trajectory. Experiment determine clearly that the parabola is the right mathematical expression, This is a conclusive proof
that defining acceleration as the change in velocity with time is the right one. Once he got the
crucial concept right he observed that the acceleration experimented by objects under the gravity
pull was the same regardless of the object (once air friction was taken out of the equation). The
independence of motion in the horizontal direction (no gravity) from motion in the vertical one, the
nature of acceleration as the time rate change of velocity resulted for Galileo in finding that there
is a privileged status of motion that was not the ”natural place” of Aristotle’s hierarchical universe:
objects upon which no ”power” was imposed, will remain in the state of motion they are: if in any
reference system they are at rest, they will remain at rest. If they are moving at a constant rate
(constant velocity) then they will remain moving at that constant velocity (straight line). Galileo
demolished the absolute universe of Aristotle (absolute space and absolute time). Space was not
absolute any more. There is a privileged system: systems from where objects either at rest or in
constant motion (non-accelerated) appear at rest or in constant motions are called inertial system:
any objected non-accelerated is an inertial system. A merry-go-round is not an inertial system.
Galileo made a gigantic jump in understanding the nature of motion. It happened 2000 years after
the greeks thought about it for the first time. But it was Ren Descartes, the great French philosopher, who for the first time clearly gave a first formulation of inertial motion. In his ”Principles of
Philosophy,” Descartes proposed three laws of nature. The first law states ”that each thing, as far as
is in its power, always remains in the same state; and that consequently, when it is once moved, it
always continues to move.” The second holds that ”all movement is, of itself, along straight lines.”
This is Newton’s first law, clearly stated in a book published in 1644.
x(t) = 10 −
3
1.2
Newton’s first law:
The First Law on inertia states that every object will remain in a state of rest or uniform motion in
a straight line unless acted upon by an external force.
1.3
Newton’s second law:
The formulation of this law was not easy. The law of inertia identified the need of a force or power
as the actions needed to stop motion, or to begin it. He said in his famous book The Principia
Mathematica:
When a force acts on an object, the object accelerates in the direction of the force. If the mass of an
object is held constant, increasing force will increase acceleration. If the force on an object remains
constant, increasing mass will decrease acceleration. In other words, force and acceleration are
directly proportional, while mass and acceleration are inversely proportional.
d~v
F~ = m
dt
(8)
This famous equation (a vector equation!) defines two physical quantities at the same time:
force and mass. What is a force?
We can determine that one is acting on a body when from an inertial system of reference (i.e.
observing from the ground) we observe a body being accelerated. We can perform experiments
and observe that when the same force is applied a body with twice the mass (amount of matter)
would have an acceleration with half the value of the other one. This mass is a measure of the
resistance to motion that the body experiments.
1.3.1
Example 1
A given force produces an acceleration of 10 m/s2 on a 1 kg mass. When applied to a second
object the acceleration doubles. What is the mass of the second object, and what is the magnitude
of the force?
F = m1 ×a1 = m2 ×a2 . The force is also F = 1kg×10m/s2 = 10 Newtons (A force of 1 Newton
will accelerate a mass of 1 kilogram 1 m/s2 . Mass m2 can be obtained then: m2 = m1 × 10
20 = 0.5
kg.
1.3.2
Example 2
A 2 kg body is at rest at t = 0. A single constant horizontal force Fx acts on the body. At t = 2s
the body has moved 5 m. Find the force. Per Newton’s second law F = ma and we know that the
acceleration is constant and lasts 2 seconds:
1
x(t) = at2 + v0 t + x0
2
4
(9)
Figure 2: An object accelerates for 2 seconds and moves a distance of 5 m.
We know that x0 = v0 = 0 so when t = 2s, x(2) = 21 a22 = 5 and a = 2.5m/s2 so F = 2 × 2.5 =
5 Newtons.
1.3.3
Example 3
A body of mass m = 0.5kg is acted upon two forces F~1 = (3, −3) and F~2 = (−2, 5). If the body
starts at rest at t = 0 find its position and velocity at t = 2s. Per Newton’s second law:
2
X
F~i = m × ~a
(10)
F~i = (3, −3) + (−2, 5) = (1, 2)
(11)
i
This means that
2
X
i
and then
2
1 X~
1
~a =
Fi =
(1, 2) = (2, 4)m/s2
m
0.5
(12)
i
And from here and from the fact that x0 = y0 = vx0 = vy0 = 0
1 2 1 2
~r(t) = (x(t), y(t)) =
ax t , ay t = (t2 , 2t2 )
2
2
(13)
When we evaluate at t = 2s we get:
~r(t = 2) = (x(t = 2), y(t = 2)) = (4m, 8m)
5
(14)
And the velocity is:
~ = (vx (t), vy (t)) = (ax t, ay t) = (2t, 4t) = (4m/s, 8m/s)
v(t)
(15)
The fact that the velocity is numerically equal to the value of the position is purely accidental and
only valid for t = 2s! The speed is v = 8.94m/s, and the direction is 63◦ from the positive side
of x.
1.4
Newton’s third law:
The third law is notable because it implies that forces act in pairs. To a force acting on a body F~
the body reacts with a force equal in magnitude and opposite in direction −F~ . Gravity is a good
example: the force of gravity occurs between two bodies: the Sun pulls the earth towards it and
the Earth pulls itself on the Sun with a force of equal magnitude but opposite direction. Immediate
question arises: why does the Earth revolve around the Sun and not the other way around? We could
use another simple example that we experience several times during our daily routines: walking.
In Figure you can see Joe walking by kicking down the Earth. The force that Joe exerts on the
Earth is the same one that the Earth exerts on Joe. But the accelerations that each one experience
are very different. Joe’s mass is mJoe = 70 kg and the Earth’s mass is mE = 5.972 × 1024 kg. We
can calculate the accelerations experienced by Joe and the Earth:
FJoe = mJoe × aJoe = FE = mE × aE
(16)
From here we see that if we estimate the force applied by Joe on the Earth to be half his weight
that would be approximately a force of 35 Newtons. Then his acceleration would be aJoe = 35
70 =
35
2
−23
2
0.5m/s while Earth’s would be aE = 1024 = 3.5 × 10 m/s . A number this small has a name:
yotta. It is a septillion times smaller than Joe’s acceleration.
Figure 3: Action and reaction make Joe walk
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