sometimes refer to this maximum possible value of static friction as
fmax s = s|N|. It opposes the component of any (otherwise net)
applied force in the plane of the surface to make the total force
component parallel to the surface zero as long as it is a
to not follow the rigorous mathematical rules that they appear to
follow as they evolve in time, for there to be a man behind the
curtain making things work out as they appear to do. Or sometimes
an entire pantheon.
Let me be therefore be precise. In the
brakes do not lock and the car is stopped with the maximum force of
static friction. In panel b) the brakes lock and the car skids to a stop,
slowed by kinetic/sliding friction.
for the feeling to back o of the brake pedal and not take full
advantage of t
whole world, the whole Universe. To the extent that this worldview is
successful, especially in a predictive sense and not just hindsight, the
physical map in your mind works well to predict the Universe you
perceive through your sensory apparatus. A perf
back, you will have a nearly perfect study guide to go over before all of
the exams and the nal. You might want to throw the quizzes and hour
exams in as well, as you get them back. Remember the immortal words
of Edmund Burke: Those who dont know history
elementary particles or are themselves composite objects36. Later in
this semester we will formally justify our ability to do this, and we will
improve on our description of things like cars and wheels and solid
composite objects by learning how they can
force. Note well that kinetic friction equals kN in magnitude, where
static friction is whatever it needs to be to hold the surfaces static up to
a maximum of sN. This is often a point of confusion for students
when they rst start to solve problems. b) Th
We can then easily determine how long a distance D is required to
make the car come to rest. We do this by nding the stopping time ts
from: vx(ts) = 0 = v0 (s,k)gts (162) or: ts = v0 (s,k)g (163) and
using it to evaluate: D(s,k) = x(ts) = 1 2 (s,k)gt2 s +
Discussion: Why dont you need to use L or v in order to nd the
tension T? Once the tension T is known, how does it constrain the rest
of your solution?
By now you should have covered, and understood, the derivation of
the True Fact that if a particle is m
This isnt a perfect example if I were doing this by hand I would
have drawn pictures to accompany, for example, Newtons second and
third law, the circular motion acceleration, and so on.
I also included more concepts than are strictly needed by the proble
would be absurdly dicult (although in principle possible) but as
long as we keep our wits about ourselves we dont have to!
100 Week 2: Newtons Laws: Continued
normal force exerted by the plane decreases (and hence the maximum
force static friction can exe
You might nd the quadratic formula useful in solving this problem.
We will be using this a lot in this course, and on a quiz or exam you
wont be given it, so be sure that you really learn it now in case you
dont know or have forgotten it. The roots of a q
N
y
fs,k
Figure 17: Block on inclined plane with both static and dynamic
friction. Note that we still use the coordinate system selected in the
version of the problem without friction, with the x-axis aligned with
the inclined plane.
use of your knowledge
H
v
0
A cannon sits on a horizontal plain. It res a cannonball of mass m at
speed v0 at an angle relative to the ground. Find:
a) The maximum height H of the cannonballs trajectory.
b) The time ta the cannonball is in the air.
c) The range R of the cannon
distorted but still recognizable form, and the constructs they introduce
to help us study dynamics still survive.
Interestingly, Newtons laws lead us to second order di erential
equations, and even quantum mechanics appears to be based on
dierential equat
actual maps were stationary, and one had to work hard to see time
on them, but nowadays nearly everybody has or at least has seen GPS
maps and video games, where things or the map coordinates
themselves move.
42 Week 1: Newtons Laws
can also be thought of
the problem. Do not do this! Solve both of these problems
algebraically and only at the very end, with the full algebraic answers
obtained and dimensionally checked, consider substituting in the
numbers where they are given to get a numerical answer. In m
itself accelerating with respect to all of the other non-accelerating
coordinate frames in which Newtons Laws hold.
Used in all problems (when I choose a coordinate system that is an
inertial reference frame).
Newtons First Law In an inertial reference f
viscous friction
turbulence
Pressure decrease
dF
v
Figure 21: A cartoon illustrating the dierential force on an object
moving through a uid. The drag force is associated with a
dierential pressure where the pressure on the side facing into the
wind of its
102 Week 2: Newtons Laws: Continued
which (if you think about it) makes both dimensional and physical
sense. In terms of the given numbers, m2 > musm1 = 4 kg is enough
so that the weight of the second mass will make the whole system
move. Note that the te
Basically, we are done. We know (or can easily compute) anything that
can be known about this system.
Example 2.1.3: Find The Minimum No-Skid Braking Distance for a
Car
One of the most important everyday applications of our knowledge of
static versus kine
also known as the orthographic projection of the object on any plane
perpendicular to the motion. For example, for a sphere of radius R,
the orthographic projection is a circle of radius R and the area A =
R2.
57Wikipedia: http:/www.wikipedia.org/wiki/Tur
his (Newtons) theory of physics was irreconcilable with that of
Aristotle, and that (since his actually worked to make precise
predictions of nearly any kind of classical motion that were in good
agreement with observation and experiments designed to test
For example units of velocity will be meters per second, units of force
will be kilogram-meters per second squared. We will often give names
to some of these combinations, such as the SI units of force:
1 Newton =
kg-m sec2
Later you may learn of other ir
have a lot of fun, I think) working on cars, jets, turbine blades, boats,
and many other things that involve the utilization or minimization of
drag forces in important parts of our society. To simplify drag forces to
where we learn to understand in gener
connected to it, although for simple problems this is not always
necessary. Either way your diagram should be clearly drawn and
labelled.
c) Choose a suitable coordinate system for the problem. This
coordinate system need not be cartesian (x,y,z). We some
through the uid. To given you an idea of how slowly a sphere
moving at 1 meter per second through water would have to be on the
order of one micron (a millionth of a meter) in size in order to
experience predominantly laminar/Stokes drag. Equivalently, a
We write Newtons second law: X x Fx = N sin + fs cos = max =
mv2 R
(165)
Xy
Fy = N cosmgfs sin = may = 0 (166)
(where so far fs is not its maximum value, it is merely whatever it
needs to be to make the car round the curve for a v presumed to be in
range)
m2
1m
A mass m1 is attached to a second mass m2 by an Acme (massless,
unstretchable) string. m1 sits on a frictionless table; m2 is hanging
over the ends of a table, suspended by the taut string from an Acme
(frictionless, massless) pulley. At time t = 0
object as the uid tends to pass over it in laminar ow. A streamlined
object will often have its total drag dominated by skin friction. A blu
object, in contrast has a comparatively large cross-sectional surface
facing forward and will usually have the tot