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you bit
very well on your first exam
i was hoping for an average of about seventy five
class average was eighty nine
so deadly business two possibilities
leader
uart very smart
is an exceptional class
order the exam
alternating
now these examples taken
three instructors
way before you took it
none of them thoughts it was too easy so i'd like to think that you are really in
exceptional class i would like to congratulate you
that you did
so well
here it is a histogram
of discourse
it's re have to decide owned this past alone
forgetting you credit for getting your homework
on this past alone
dividing line between fact and fail would be
sixty-five
that means
michael sends of the class
would fail
unusually low normally that is around
fifteen percent
what time will tell whether you are indeed exceptionally smarts
or whether the exam
mostly the good news also is
two pieces of good news
that we promise that the books will arrive
at the coup
today we're going to talk about
springs about penguins and about
simple moment also made as one of the king
topics in
ada wong
if i have a spring
ended the relaxed length of the spring
spring
ecole de tactical zero
and i are extent the string displaying zippy
then there is a force
wants to drive
this spring back
equilibrium
and it isn't
extreme mental facts
that many strengths we call them ideal springs
for many springs this force
proportional
to the displacements
acts
so this is acts
if you make x three times larger
that restoring force
is three times larger
this is a one-dimensional problem told to avoid a vector notation
we can't simply say that the force
therefore is minus a certain constant which we called the spring constants
this is called the spring constants
and the spring constant
has units milken's reader
one of my time
takes care
off the
direction
annexes positive
then the forces in the negative direction when at his negative
reported in the positive direction
it isn't
restoring
whenever there's really no relation between f
annex holds
that is referred to
as one claw
how can we
measure the
sprinkle instant that actually not too difficult
i can use gravity
here is the spring
in its relaxed situation
i hang on the spring
a mass and
and i made use of the fact
regret it you know
exerts a force on that
when you're back in new equilibrium this is the new equilibrium position
then this spring force
of course must be exactly the same is angie
no acceleration when the thing is addressed
and so i could not make fault
whereby
i could have he acts
and i could have here
this course at which i know because i know the masses i can't change the
masses i can go through a whole lot of them
you'll see data points which scattered around
straight line
andy
spring constant follows then
if you take
if you call these belfast
you call this delta x
and the spring constant k
delta f
bye bye
to act as you can even measure if you don't have to start necessarilly
at this point british printers relaxed you could already start when the spring
is ready
under tension that is
problem
you'll be surprised how many springs really behave very nicely
according to hoops law
and i have one year that's not a very expensive
spring issue here
and it is here
holder
onto it so it's ready a little bit on the stress
at doesn't make any difference
these smart here at fifteen centimetres apart and every kind of that killed one
kilogram ong
you will see
that it's goes down by roughly
fifteen centimetres
goes on to this mark
i put another killer donald
goes down to just mark
i put another kilogram loans
and it goes back to get more
all the way down
i did i think the wall also what i've done a side effect if we went along his
curls
if i take them off
it it is an ideal spring
it goes back
which originally
which it does
this requirement of course for an ideal spring
it behaves
according to hoop law
i can of course overdo things
i can take a spring like this one
and structure to the points that is no longer behaves
hoops lol
i condemn acute
uh... i can do prevalent
the formation
look that's easy
for sure who flow is no longer
look how much longer this spring is and it was before
soda comes of course the limits
how far you can go
before
you permanently deformed
what i have done
now we've got spring probably beginning i went up along a straight line
and then
something like this must have happened
i got a huge extension
my fourth did not increase very much
and then when i relaxed when i took my four self
the spring live longer at the end
that it was at the beginning
so i haven't next extension ritual will be there
and that's not very nice of course to do that
so we'll talk holds only within certain limitations you have to
uh... will be a certain amount of
discipline
there are race that you can also
measure the spring constants in a dynamic way
which is actually very interesting
uh... i have here
the spring
and this spring with his ex equal zero
and i've attached now to the spring
and max m
it has to be on a
perch on the surface
and you will see when i extended over the distant x
that you get your
force
you spring forces
that brought you back
we have of course gravity and g
and we have the normal force from the surface so it is in the in the white
election
there is no acceleration
so i don't have to worry about the forces in the right of action
if i get this thing also late
i know that i really think it will start to also make about this point back and
forth
then as i will show you know
you will find that the purity of of also nation
time for one whole solution
is to pipe
front disc railroads
of the mass and
divided by the spring constant k
out of the right that you will see that shortly and i will give you measured the
periods
and uh... you know the mass
then you can
calculate k
alternatively if you knew cake
and you may get a few minutes
you can calculate the maps
even in the absence of gratitude i don't use credit year to spread all over the
allowed you to measure
uh... a mess
unity absence of
periods that you see
the time that it takes for this object to also made wants
back-and-forth
is completely independent of how far i move it out which is very known
intuitive but you will see the death comes out of the derivation
there is no dependents
on how far i move it out
soweto i also made it like this
what i also labeled like this
as long as groups law holds
you will see that the purity of his independence of what we call that
and
so i'm going to provide the situation now for an ideal case ideal case means
controllers holds
there's no friction
throwing itself
has negligible mass compared to this one
let's call it the massless
so i'm going to ride on
newton's second law
and they which is all in the extent action
equals minus k x
aid the second derivative
of position
for which i will write
x double dot
and x double box one dot is the president of the debate velocity two
adopted the acceleration
plus que x
equal zero i divide by m and i got
x double dot
was ok over m
arms x equal zero
and this is arguably
the most important
equation in bold
of physics
two different equation
some of you may already have sold differential equations
the outcome of this you will see he's very simple
axes of course changing in some way
as a function of time
and when you have the correct solution
forex as a function of time you substitute that back into
that differential equation
that equation
we'll have to be satisfied
walk towards a solution be
to this differential equation
i'm going to make u
c this also nation first i'm going to make you see
acts as a function
and i'm going to go to the following way
i refuse spray paint can
which is
suspended between
through springs
and i can also made it
particularly which is you extend action
x changes with time
the climax is idle introduce
morning on this spring
when the spray paint is going to
spray i'm going to pull next spring if i can do it at a constant speed
then you get horizontally a time axis and of course brutally
over a particular you will get the position of x
so i wanted you just see qualitatively
what kind of who we have to curb
act as a function of time is
which i will have to satisfy the differential equation
alright it's always a messy experiment because
pages dripping
but i will
tried to
at this briefing
going there we go
of paul
you give me hands
now could you please
i'll cut it here and then you
very careful because it's it is messy
or let's put it did not take it out this way
before back
useful yeah great
i hold up the talks with the conceded point
okay
what i would remind you off
finest words reminds me of course trying to attack
sinusoidal record time
well let's try it
to substituting that equation
scientist or it's or a post on solution
whichever one you prefer it makes no difference
so i'm going to substitute
in this equation this my trial function
that acts as a function of time
is a constant eight i will get back to that in a minute and cosign on the
campaign
looks fine
there's eight we call the amplitude
nor did the coastline from tunis the highest values plus mum
and the lowest values might as well
the amplitudes
indicates that is the part of his placement from zero on this side would
be plus a
and on the side would be minus a
for that to me this
there's only got
recall be angular
we quincy
don't confuse it with angular velocity
record angular frequency and really try to say
units are ingredients
four seconds
singers and velocity
if i a bump this time a little teeny
if i had bombs that
two pi
divided by omega
if i could bump is fine by two pi divided by only got then this angle here
increases by two pi radiance
which is three hundred sixty degrees and so that's the puntodat date
forty also nation to repeat itself
so this is the period of the oscillation
and that is in seconds
and you can
determine if you want to you can define the frequency of the ulcerations
which is one overtly week we express always in terms of hurts
and then you know we have what we call
if anything go
i will return to that that's in radians
file function i'm going to substitute now
into this equation
thing i have to do i have to
find what the
second the ridiculous
all x as a function
well that's my function
i have here
bursting
bertha riveted x dot
that becomes minus aid
omega i get unlimited out because there's a pioneer now i have to take the
derivative of the function
itself
so i get the sign
only got killed
spock
because i could have started off he was assigned provide hope you realize that i
just picked the coastline
x double dot
now i got another will make out
so i get mine is eight
omega script the derivative of the sign is the co-sign coastline
omega keep
was fired
and that is also minus
omega square
contact because notice i have eight
goldstein told me that keep us fight
which itself is x
so now i'm ready to substitute this results into that differential equations
this muscle always hold
for any belly over explaining moment in time
and therefore the only way that this can work is evolving as korea is ko for him
only guess career
must be
and therefore we now have the solution
through this problem
so we have
omega
equals the square root
okay over and and the periods is to fight unto scrambled
and
okay
and what is this is a striking really remarkable
that this is an independent all the
and thank you
and it's also independent all this
endl
five his face angle what is this business of the space angle it's a
peculiar thing that we have there
well you can think about the physics actually
when i started also nation
i have two choices
of two things hike and started off at a certain position which i can choose
i can give it a certain displacement from general and simply let it go
but i can also when i let it go give a disturbing velocity that's my choice so
i have two choices where i let it go
and what philosophy i did
and that is reflected in my solution
namely
that ultimately in the solution
i get the result of me and the result of five
which doesn't determine the periods
its results from what we call might initial
conditions
and i want to
do an example
whereby you see how a_d_m_ fired
immediately follow
from the initial conditions
so his example
i really see objects
exit polls veroa akito zero so i really think etc equilibrium
at that moment in time
i give it a velocity
which is minus three meters per second unit so listened and p_s_ units
the spring constant k
people spend millions from the rest and the mass of the object
is all point want him
and now i can ask you
what now is acts as a function of cotton including the amplitude a including the
face angle
at first calculate almeida
that is the screwed ko grandpa
that will be ten radians per second
the period three
which is to apply divided by omega
would be a roughly
six points
to eight seconds
and the frequency f
would be about
all-points one six hurts
just to get
some numbers
one point six to three
this is not mine
babies is all points
six to eight
and this is lock horns
to pipe
divided by the major you can see missus ten six divided by ten is about all
point six
all right so now i do know that people zero
x equal zero
so i see my solution right there
right here
according to equal zero
and our taxes zero
so i get zero
equals age
onto the coastline
they did not zero
if i really sitting equilibrium and i give it a velocity of three meters per
second is going to also later april zero
so he only solution is a close on five zero installed at least nearest
five by over two
applied
three part five three five
the only two possibilities
now i go to my next
initial condition that the velocities minus three
so here you see the equation for the velocity
this is minus three
at people zero
so minor three
equals
might is a d_n_a_ is
we don't know yet
pregnancy
and there we have only discredited
omigod story which is ten
d a zero i get to sign a fired
if i could quiet over too
and the final fight is one
and so you find immediately
that eight equals plus oprah winfrey
and sort of solution now
which includes now apply in haiti
is that x
equals plus the whole point three gp
fun to co-sign
omigod
which is fenty
what are
over too
so you see that the
initial conditions
what the conditions i people zero they determine my eight they determine my
facing
if you have children
this as the phase angle three by overthrew
that would be fine
you would have found minus nine year and that's exactly the same
so you were found
i want to demonstrate to you that d
punitive also nations moving toward it as that may be
is independent
all the
and two
that i give the object
and i want to do that here was this at frank
i have a an object here
is objects
has amassed
another eighty
six plaza minus one gram
for them on
i'm going to also made it
and we're going to measure the periods
but instead of measuring long periods
i'm going to measure
ten periods
because that gives me a smaller
uncertainty a small relative ever in my measurements
so i'm going to do it as an amplitude
which is fifteen centimetres let's make it twenty
animators
so i get ten t
i got a certain number
and i got an error which is my reaction ever
which is about a tenth of a second
that's about the reaction there were that we all have roughly
then i will do it at forty five minutes
we've got a ten t
and we get
again
close minus one point four point one second
penalty
how much they differ
they should be saying it isn't ideal spring
within the uncertainty of my measurements
you see the timing there
i'm going to give it a try sending me the whole separatists
and then i will started when it comes back to you so i will allow it while
also nations first
that's an easy for me to see it stands still
when i started
when we see
fifty point one six
fifteen
point one six seconds but are you going to write this spring collection from
this now
as you know to mass
uh... you know the time
now i'm going to be with him
displacement and amplitudes
which is twice as high
trying to get forty centimeters
loses ten
forty centimeters a huge displacement
fifteen point one three
fantastic agreement within the uncertainty of my measurements
there within three hundred of a second
before that you tried many times
you will get that close because
my reaction times really not much better than a tenth of a second
i will tell you something else
which is quite interesting and that is
how the behavior of the peer reviews
on the
on the mass of the objects
i have your another caller
which raised roughly the same
i'm going to add to the tour together
and so we get and two
is about three hundred seventy two
bottom line is one gram the puzzle miners want comes in because our scale
is no more accurate and won't grant so we put them both on the scale
and we find this
the uncertainty
so now
i'm going to measure depend periods
office
object which mascot and tour took twice the mass
so that should be
this croat
of and to divided by anyone
times ten times
the periods
of and one
and so i can make
a prediction because this is the square root of two
and i know what it is
so i will take my calculator
and i'll take this crew to film
and multiply that by
let's take
fifteen point one five
and so death comes out to be twenty one point four two
twenty one point four two it's not clear that these two was meaningful and i'll
come to sixty fourth all the question
one of the uncertainty is a prediction
and economic growth and intricate
so what i'm telling you know me confusion a bit not meant to be which i
really won't hold you responsible for it
you may not think
that the insert into in these measurements
photos from the uncertainty in this which is true which is about all point
six percent
and from the uncertainty in this
so this has about an uncertainty all point six percent
about as well because i'm at work and also a huge issue
since it is only one out of the fifty which is low
you may think that the uncertainty in there
equals the ***
of three hundred seventy proposal might as well
provided by home of eighty six plus or minus one
and now you may argue and that's completely reasonable that you would
argue that way
you would say well this is roughly a quarter of a percent guarantee on the
desk railroads
and this is roughly
half a percent ever one out of two others about pops
so you would have agreed to a rust quarter plus optics about all point
seven and because of this groups
that becomes all point three five percent and that's wrong
and the reason why that is completely wrong
that has to do is the fact
these two errors are coupled to each other
that one of the eighty second including the three seventy two
the best way i can show you this suppose i measure and long divided by m wuwm
which would be one of the navy six bottom line is one divided by home of
the eighty-six plaza minus one
that number is one with a hundred miro's his number is one you've accepted by
half of one object to divide by the same object
rose if you would say i love it is on top of the sender
and it is a opera sender you would say to make your has an air of one percent
that's not the case
so i will not bother you with that i will not hold you responsible for that
but it turns out that if you
which correctly
and you take the ever
of this into account of about opal six percent
that the ***
ratio is really much less than point two percent you can almost forget about it
idle allowed generously
four one percent ever in the final answer
and so
i stick to my prediction that the ten th
kal double the mass
is going to be
and now we're going to get the observation
ten p
and tool
which is double the mass
and that of course always has mine
uncertainty of my reaction time
nothing i can do about that
and we will compare these two numbers
so i will put the the automatic top of it
was here
anything together so that they want
we go up
overdid it correctly this crowd of two
fifteen point one five
well here today
amplitudes something like the maybe thirty five senators
here we go up
to much slower you see that
nine i'm not looking
twenty one point three six
twenty one point
three sixty
and rounded off for if you want to
and use it if you agree with the spectacular
within the uncertainty over my measurements
amazingly well you've got every move this to of course because if you have an
uncertainty of point two hits a little silly have difficult to hang in there
do you see
attendee this spring is very close to an ideal spring
its full days weeks will
and it is also
actress
here is the pendulum
here's the mass
and uh... i it's offset
and then go figure
lengths of the pendulum isabelle
length of the street
there's gravity here
and g
and the auto force on the
object
oneonta force
is the tension
don't confuse that was purely fetuses pension symptoms
but only two forces there's nothing helps
the thing is going to are corrupt
like this and going to also make
as well
accordance whatever action
and i called it
the excel rection
equals zero
well i'm going to decompose the
tension into
the wired into the
x election as we have
done before
this is going to be the why components
x components
so it is my components
equals t
co-sign feder
the x_ components
equals p
i think that
and now i'm going to write down the differential equations of motion
the expectation
not
second
milton second lol
anne
equals
this is the only fourteen d_x_ direction
it's a restoring force just like with the spring
therefore have to get it
minus sign
equals minus
the sign fade out
details could be easily be a functional theta so i have to allow for that
the sign of sadar
equals expected to get position x
divided by l
so can i can write for these
minus t
which may be a functional sadar
contacts
divide by l
that it might differential
equation in the extent rection and i prefer
always for the city
write-down acts double-talk
alkali direction
in light of excitement
m
why double dot
equals
this is my plus direction so i have
he goldstein potato
minus angie
the equation one
and inclusion too
as of now we have to sold
to couples differential equations
hopeless fast it looks like a zoo
and it is is it
and now we're going to make some approximations
andy approximation that he will make which you all can see
in physics
when something also mates
is what we call
the small angle
approximations
small angle we will not allow fade out
to be calm
too large
i'll be qualitative
what i mean by too large
wednesday dot recusing radiant equals much much less than one
recall that
a small angle
if that's the case because i know faye dot
is very close to one
you will say well hobart apply how close to one ok five degrees
because finding job board nine
nine six
that's close to what
ten degrees
the coastline is all-important ninth h five
but only one on the hapless and different
full-blown
even attend agrees you doing extremely well
so there's this
consequent number one
of this morning approximation
but there is a second consequence of this morning approximation
mocha activity excursion that they'd all get make from equilibrium
inaccurate should that there's a big
look at the excursion it makes in the white election if this is small
wary smaller
then the excursion in the extraction provided
that you angle is small
i'll give you an example
at five degrees
this excursion
is only four percent
of this explosion
attend agrees
there's excursions only nine percent
will be excursion
and since the excursion in the white arctic chin is so much smaller than the
actual rection
restate that the acceleration in the right of action candy approximated
roughly zero
there's almost no acceleration
the right direction
with these two
conclusions which follow from the small angle approximation i go back
to my equation on the tube
and i find that zero
equals
which could be a function of data
co-sign state as well
minus
and g
so i find that to me
equals and g
is no longer even a functional state a
so i simply have to give my small and an approximation
that i can
victory
the same angie it's approximately i still political side
i substitute is backed up in my equation
number one
and so now i get
and
and x double dot
and i'll bring these on the other side
plus
teams now and g
and g
fund x divided by l
equal zero
and all
wonderful result x noble dot
close geo file contacts
equals zero
and this is such a beautiful results
that almost
makes me cry
this
s a simple how moment also mason
this equation
like carbon copy
off the wild that we have there
here we have ko brad
and bear
we have g over l
that's all
other than that
there is no difference
so you can write down immediately dissolution
through this differential equation
x
will be some amplitudes
five the co-sign only got to you
was fired just as we had before
omega
will now be discouraged
also jeannie over al and sort of period of the pendulum
will be too quiet
and his crew
apple l
o_j_
just falls into our lab because we did all the work
of what you do realize that these
results for pendulum
have their restrictions
small angles and we discussed qualitatively
how small you would like to allow
and also
the mass has to be exclusively eight-year annul into straight recall
that
en masse was
to give you some rough idea
what these periods will be
substitutes
for l one meter
are you take four g_ nine point eight
that this growth
people to fly by two five
and what you find it
the period is about two seconds
so the pendulum mummy too long
as a period of about two seconds
so to go from here
subodh won't act
if i make it four times shorter
l four times photodisc roads before it's true
then the periods
the period is changing four times shorter the period would be to country
make a roughly twenty five centimeters
i'm not trying to be very qualitative year laudable period must be about one
seconds
x roughly one sec
so you see
that's the
periods is extremely sensitive
i know i want to
compare with you with the results that we have
forty spring
with results that we have
that pendulum
forgive you
some further insight
we have the strength
and we have the pendulum
and i'm only going to look at the periods
which here is to find
divided by the screw
emal okay
and here is to apply
and his crew l
if i don't hear
there is a massive
if i look here
independent of the mass
why is there a mass in here
that it's very easy to see
if i think a spring
and i extended spring over a certain distance
then there is a certain force
that i feel
but fourteen independent of the maps that i put at the end of this point
this plane doesn't know what the masses you going to put on
all that noses into long and i want to go back to equilibrium
that forces affixed force it by double the mass
that's fixed force will give on double the mass hockey acceleration
if the acceleration goes down
the period of automation goes up is very clear
so you can immediately see
that was just plain the basque must enter into the periods
now go to the pendulum
if i've doubled the mass of my bald at the end of the pendulum
vertical component of the tension will also double
that means this restoring force which is proportional with attention will also
double
for now to restoring force doubles and the maya doubles the acceleration
remains the same
the period remains the same
so you can simply argue
that there should be no glass in here and there isn't
how about these k
if gays height
then a spring istic
or that meets tapes bring it means
that if i give it thus more extension
that this bring forth hilts
notes reports
acceleration on a given mass will be higher
if i have a high acceleration the period will be short and that's exactly what
you see
he's getting his height
it period
will be short
imagine that you have a pending the outer space there's no gravity nothing
the pendulum will not swing
the period of the pendulum
will be infinitely long
going to the shuttle
we had a perceived gravity
in their frame of reference
receive their way please remember
they're perceived gravity is zero you take a pending in the shuttle people
either this angle you let it go it will stay there forever and ever and ever
the period is infinitely lol
but take a spring in the shuttle
and that the spring also made
and it does
so you can actually measure the mass of an object using the spring on the
shuttle and that also maybe if you know to sprinkle instant and that's the way
check should be done
your cv
that these
things make sense
think about it
in a rational rate
we have here
in twenty six one hundred
the mother of all pendants
it is a pendulum
it is a pendulum which is something
five point one leaders
long
there's a mass at the end of it
which is fine fifteen kilograms
five-point won't hate needles
and uncertainties about five centimeters
we can measure it
any better
and the math at the end of its which doesn't enter into the periods
is about fifteen kilograms
the periods
which is to pipe
crimes this croat apple ella wenjie
if you're substituting
your length of five point one meters
you'll find four point
seven seconds
quarter-point
fight stick
since you have a lot of the sender aaron l
you're going about hahaha sent ever in europe iriarte
so that is about
oprah winfrey
seconds
so this is mike prediction
and now i'm going to
also waited for you
and i'm going to do it
from two oh
different angles
i'm going to do it at a
five degree angle and i'm going to do it at eight
ten degree angle
in order to get my
well if you've ever down i will also make
ten times
so i'm going to get
at an angle data maximum
of roughly five degrees
fenty
equal something plus or minus my reaction time
which is all point one per second
and then i'll do it from ten degrees
and how do we can't enti
and again my reaction time
is not much better all point one second
but i do that for us
of a movie about the way
because of that
fifteen kilograms object
this that is not
all right
by the mark you on the floor this is about
five degrees
there's about ten degrees
affair of wood from five degrees
i will let it swing one also mission when it comes to a whole p_l_o_ start
the time of the twenty d_e_c_'s
account on you
when it comes to
ready you sure
i'm ready to
okie cokie men don't confuse me again now
you're completely responsible for the company
so we don't have to tell me is when they went eight or nine is coming out that's
all i want to know
don't even bother me with three don't even bother me the sport
just let me know when i have to
getting position for it
the final q
not it is almost no dumping on the spending the entry troops remains almost
insane reserves the
vfx you could actually see that there was a ready some kind of friction post
let me know
nine right
forty five
point seven zero
forty five point seven zero one of my chalk
forty five
point seven zero
what what my prediction
yeah
yeah exactly
you get the picture
that is pure luck
because my accuracy is no better than a tenth of a
tecnologia form ten
ten degrees
and i want to show you know that
effect on the angle of the orphan five to ten
small
so small that you cannot measure it was in the accuracy of you mentioned
began
relaxed and calm
uh... nerve-wracking
fairly well
janet
security regretted the help
forty five
point seven point
one of the most remarkable things i just mentioned to you
is that v
period all the ulcerations
is independent
of the mass
of the object
that would mean
if i joined the ball
and i think that was the ball
that you should get that same period
or should you not
i'm asking you a question before we do this
awful experiments
what the period come out to be the same or not
some of you think it's the same
have you thought about it
that i'm a little bit volunteers albrecht and that therefore
may be effectively deter that length of the string
has become a little less
if i set up like this and if the length of the spring is a little less
the period
would be a little shorter
be prepared for that
we are and i'm also prit
well i'm not quite prepared for me
i don't try
to hold my body of arizona qualified possibly can
in order to
be at the same level as the ball
i will start
when i come to a whole peer there we go
you'll go
that's right
god
the
all
it
here
hack
attitude
with walter willett
forty five
quantic mother wanted all cornwall second physics worksite telling you
alkenyl monday have a good weekend