Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
all of the stuff with a v in it to the left, multiply the dt to the
right, integrate, solve for v(t), set the constant of integration,
and answer the questions. Ill do the first few steps in this for
you, getting you set up with a definite integral: dv v
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
velocity at the top is vt, down. In order to stop you before y = 0
(the ground) you have to have a net acceleration a such that:
v(tg) = 0 = vt atg (179) y(tg) = 0 = H vttg 1 2 at2 g (180) If
we solve the first equation for tg (something we have done
many
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
how to solve the equations of motion for at least the two
limiting (and common) cases of Stokes and turbulent drag. The
entire complicated set of drag formulas above can be reduced
to the following rule of thumb that applies to objects of
waterlike densi
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
very hard to actually find and solve the equations of motion for
a streamlined object that falls subject to a Stokes drag force.
We begin by writing the total force equation for an object
falling down subject to nearEarth gravity and Stokes drag, with
do
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
experiences driving cars and the like. So much for description;
what about dynamics? If we differentiate this equation twice,
we get: d~x dt = d~x dt + ~vframe (198) d 2~x dt2 = d 2~x dt2
(199) (where we use the fact that the velocity ~vframe is a
constan
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
numbers into a calculator correctly. This course is intended to
teach you how to correctly obtain the algebraic expression that
you need to numerically evaluate, not drill you in calculator
skills52 . We start by noting that, like Atwoods Machine and
one
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
but even if it does the answer is relative to another inertial
reference frame which begs the question, a very bad thing to
do when constructing a consistent physical theory. To avoid this,
an inertial reference frame may be defined to be any frame
where
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
the week that we cover fluids, when an object is sitting at rest
in a fluid at rest with a uniform temperature, pressure and
density, the fluid around it presses on it, on average, equally on
all sides54 . Basically, the molecules of the fluid on one side
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
forces perpendicular to the gravitational field and pressure
gradient still cancel even then. 55Wikipedia:
http:/www.wikipedia.org/wiki/Drag (physics). This is a nice
summary and well worth at least glancing at to take note of the
figure at the top illust
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
pretty much ignore this transition. It is just too damned difficult
for us to mess with, although you should certainly be aware
that it is there. You can see that in our actual expression for the
drag force above, as promised, we have simplified things ev
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
psychologically and occasionally computationally useful to do
so. Psychologically because they describe what we experience
in such a frame; computationally because we live in a noninertial frame (the surface of the rotating earth) and for certain
problems
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
dimensionless constant characteristic of the two surfaces in
contact, and N is the normal force. c) The direction of static
friction is parallel to the surfaces in contact and opposes the
component of the difference between the total force acting on
the o
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
physics should just ignore this adverb. Students going on in
physics should be aware that the real, relativistic Universe those
times might not agree. Week 2: Newtons Laws: Continued 119
Displacing the origin is described by: ~x = ~x ~x0 (195) and as
abov
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
coefficient of kinetic friction, a dimensionless constant
characteristic of the two surfaces in contact, and N is the
normal force. Note well that kinetic friction equals kN in
magnitude, where static friction is whatever it needs to be to
hold the surfac
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
cos = mv2 R (168) and finally solve for fs: fs = mv2 R mg tan
sin tan + cos (169) From this we see that if mv2 R > mg
tan (170) 106 Week 2: Newtons Laws: Continued or v 2 Rg >
tan (171) then fs is positive (down the incline), otherwise it is
negative (up
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
the first week/chapter. For example, we can integrate twice and
find vx(t) and x(t), use the latter to find the time it takes to
reach the bottom, and substitute that time into the former to
find the speed at the bottom of the incline. Try this on your
ow
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
possible distance. We achieve this by answer the following
questions: Find the minimum braking distance of a car travelling
at speed v0 30 m/sec running on tires with s = 0.5 and k =
0.3: a) equipped with ABS such that the tires do not skid, but
rather ro
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
neglecting certain wellknown and important facts or forces
that appear in realworld problems in order to concentrate on
ideal problems that illustrate the methods simply. It is time to
restore some of the complexity to the problems we solve. The
first t
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
explains a lot of the things we skim over below, at least in the
various links you can follow if you are particularly interested.
Week 2: Newtons Laws: Continued 97 a) We will only consider
smooth, uniform, nonreactive surfaces of convex, bluff objects
(l
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
press conference, you need a haystack or palette at the
mattress factory or thick pine forest that will uniformly slow you
over something like 10 or more meters. I myself would prefer a
stack of pillows at least 40 meters high. but then I have been
known
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
and formal algebraic reasoning for you, and are only included
from time to time to give you a feel for what reasonable
numbers are for describing everyday things. 102 Week 2:
Newtons Laws: Continued which (if you think about it) makes
both dimensional and
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
in this frame that can also be solved like Newtons Second Law
in the (accelerating) frame coordinates. That is, we would like to
write: F~ = m~a (205) If we compare these last two equations,
we see that this is possible if and only if: F~ = F~ m~aframe =
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
through a fluid experiences a drag force of Fd = btv 2 . In the
figure above (generated using the numbers given in the ram
example), m = 100 kg, g = 9.8 m/sec2 , and bt = 0.392, so that
terminal velocity is 50 m/sec. Note that the initial acceleration
is
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
problems of this sort. Example 2.2.3: Dropping the Ram The
UNC ram, a wooly beast of mass Mram is carried by some
naughty (but intellectually curious) Duke students up in a
helicopter to a height H and is thrown out. On the ground
below a student armed wi
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
problem! A car of mass m is rounding a circular curve of radius
R banked at an angle relative to the horizontal. The car is
travelling at speed v (say, into the page in figure 20 above). The
coefficient of static friction between the cars tires and the ro
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
versa. From above, we know that I = ML2/3 for a rod pivoted
about one end, therefore: = MgL 2 sin() = ML2 3 = I or:
= d 2 dt2 = 3g 2L sin() (499) independent of the mass! 5.6:
Solving Newtons Second Law Problems Involving Rolling One of
the most common a
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
stopping distances for ABSequipped cars are some 18 to 35%
shorter than nonABS equipped cars, for all but the most skilled
drivers (who still find it difficult to actually beat ABS stopping
distances but who can equal them). One small part of the
reason
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
x(t), if we care to, but in this class we will usually stop here as
x(t) has pieces that are both linear and exponential in t and isnt
as pretty as v(t) is. 62Just kidding! I know you (probably) have
no idea how to do this. Thats why youre taking this cou
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
air or water do not just keep speeding up ad infinitum. When
they are dropped from rest, at first Week 2: Newtons Laws:
Continued 111 their speed is very low and drag forces may well
be negligible58. The gravitational force accelerates them
downward and t
Rehman College of Rehabilitation Science Hayatabad, Peshawar (Dr of Physical Therpay 5 Years)
economics
ECON 103

Spring 2016
we can change clocks at will when considering a particular
problem by means of the transformation: t = t t0 (192)
where t0 is the time in our old timecoordinate frame that we
wish to be zero in our new, primed frame. This is basically a
linear change of