Julius Sumner Miller: Lesson 7 - The Simple Pendulum and Other Oscillating Things

00:14:49
https://www.youtube.com/watch?v=xr4EtSz7zek

概要

TLDRIn this video, Julius Sumner Miller delves into the captivating world of pendulums and oscillating systems. Starting with the simple pendulum, he explains how its period depends solely on its length and is independent of the bob's mass. He demonstrates this by oscillating pendulums of three different lengths and noting that their periods relate to the square root of their respective lengths. Miller also introduces the concept of a physical pendulum using rods, hoops, and discs, explaining how they can be equated to simple pendulums in terms of period. He illustrates this with examples, showing that these objects have equivalent simple pendulum lengths that match their periods. Moving on to springs, Miller displays the oscillation behavior of springs with different loads and lengths, urging viewers to experiment and observe for themselves. He concludes with a fascinating demonstration of coupled pendulums, where the motion transfers between them, highlighting principles relevant to electric circuits. Miller's presentation not only provides educational insights but also encourages curiosity and hands-on exploration of physics.

収穫

  • 🎢 Simple pendulums' periods depend on length, not mass.
  • 🔍 The period is proportional to the square root of the pendulum length.
  • 🧩 Physical pendulums have equivalent simple pendulums.
  • 🔗 Coupled pendulums transfer motion between each other.
  • 📏 Springs exhibit unique oscillatory behaviors by length and load.
  • 🌍 Period varies with gravitational location (e.g., Earth vs Moon).
  • 🔬 Experimentation reveals physics principles.
  • 🛠 Pendulum experiments encourage learner exploration.
  • 🔄 Mechanical issues prompt adaptive experimentation.
  • 👨‍🔬 Curiosity is key to uncovering physics beauty.

タイムライン

  • 00:00:00 - 00:05:00

    Julia Sumner Miller introduces the topic of pendulums, describing the motion of simple pendulums using an idealized model comprising a massless string and a bob. She explores the period of a pendulum, which is dependent on the length of the string and the gravitational constant. Demonstrations include using pendulums of different lengths, revealing that the time period is proportional to the square root of the length. This relationship is shown through specific experiment results demonstrating different oscillation times, highlighting the formula's validity. Miller emphasizes the independence of pendulum period from the mass of the bob by comparing various weighted pendulums of equal length, all having identical periods. She further contrasts simple pendulums with more complex "physical" pendulums, exemplifying with rods and hoops, illustrating how their equivalent simple pendulums share the same period based on specific proportional lengths.

  • 00:05:00 - 00:14:49

    Miller details an exploration of physical pendulums using various shapes like rods, hoops, and disks, each time explaining how an equivalent simple pendulum can be modeled with the same oscillation period. She discusses the peculiar properties of these pendulums showcasing the mechanical beauty in their oscillation. Additionally, she introduces oscillating springs as another form of periodic motion, prompting curiosity by leaving the audience to ponder how spring length affects the motion. Coupled pendulums are demonstrated to exhibit energy transfer between them, analogous to behaviors observed in electric circuits, thus broadening the principle's relevance. The discussion ends with Miller urging the audience to explore oscillating systems, invoking appreciation for the underlying principles of physics.

マインドマップ

Mind Map

よくある質問

  • What is discussed in this video?

    The physics and properties of simple pendulums and other oscillating objects.

  • What determines the period of a simple pendulum?

    The period is proportional to the square root of the length of the pendulum.

  • Does the mass of the pendulum bob affect the period?

    No, the period of a simple pendulum is independent of the mass of the bob.

  • What is a physical pendulum?

    A physical pendulum is any oscillating body that is not a simple pendulum, such as rods, hoops, and discs.

  • How does the period of a physical pendulum compare to a simple pendulum?

    A physical pendulum has an equivalent simple pendulum length that matches its oscillation period.

  • How is a pendulum with different materials used in the demonstration?

    Pendulums made of brass, aluminum, and cork are used to show that period depends only on length, not material.

  • What is unique about the motion of coupled pendulums?

    Energy transfers between the pendulums, causing one to stop while the other moves, demonstrating coupled harmonic oscillators.

  • Does the length of a spring affect its oscillation?

    Yes, the period of a spring's oscillation depends on its length and stiffness. Exploring two springs offers additional insights.

  • How can physical pendulums be used?

    Physical pendulums, like rods and hoops, illustrate properties of oscillation and equivalent simple pendulums.

  • What encourages further exploration of these concepts?

    Julius Sumner Miller encourages curiosity and experimentation to discover more about pendulum and oscillatory motion.

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  • 00:00:00
    [Music]
  • 00:00:16
    how do you do ladies and gentlemen and
  • 00:00:18
    boys and girls
  • 00:00:20
    i'm julia sumner miller and physics is
  • 00:00:22
    my business
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    and our very special business today has
  • 00:00:26
    uncommon enchantment because the motion
  • 00:00:29
    is beautiful to witness
  • 00:00:31
    and wonderful to understand
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    we are going to talk about simple
  • 00:00:35
    pendulums and other oscillating things
  • 00:00:39
    consider the following
  • 00:00:41
    if we have a rigid support and there are
  • 00:00:44
    none really because everyone shakes
  • 00:00:46
    however rigid
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    and we hang an ideal string by an ideal
  • 00:00:51
    string i mean a string that has no mass
  • 00:00:53
    no weight has no inertia has no tension
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    is perfectly inelastic ideal
  • 00:00:59
    mathematically and we hang a small bob
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    on the end
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    and we displace it from this equilibrium
  • 00:01:06
    position and let it go
  • 00:01:09
    it oscillates as a simple pendulum
  • 00:01:13
    now we wish to explore its motion
  • 00:01:17
    when you study the lesson in this
  • 00:01:19
    business
  • 00:01:20
    you will find that
  • 00:01:22
    the period which is the time for a
  • 00:01:24
    complete trip comes out to be
  • 00:01:27
    some such expression with which i will
  • 00:01:30
    not have much to say about which i will
  • 00:01:33
    not have much to say
  • 00:01:34
    it's governed as we see by the square
  • 00:01:37
    root of the length and depends upon
  • 00:01:39
    where we time it
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    you know for example the g is one thing
  • 00:01:43
    on the earth and quite another thing on
  • 00:01:45
    the moon
  • 00:01:46
    g is 32 feet per second per second on
  • 00:01:49
    the earth but about one-sixth of that on
  • 00:01:52
    the moon
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    as an incidental fact matter you would
  • 00:01:56
    weigh therefore one-sixth as much on the
  • 00:01:58
    moon as you weigh on the earth
  • 00:02:00
    now i wish to do another pretty
  • 00:02:03
    experiment
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    with several
  • 00:02:06
    pendulums and i suppose maybe we could
  • 00:02:09
    say pendulum this may be the plural of
  • 00:02:12
    pendulum now what am i going to do
  • 00:02:15
    i am going to hang up three of them as i
  • 00:02:18
    have them here soon to witness
  • 00:02:21
    one
  • 00:02:22
    two
  • 00:02:24
    three
  • 00:02:25
    and how long are they going to be this
  • 00:02:27
    one is going to be 10 centimeters long
  • 00:02:30
    and this one 40 centimeters long and
  • 00:02:32
    this one 90 centimeters long so the
  • 00:02:35
    lengths are 10
  • 00:02:37
    40
  • 00:02:38
    90 centimeters
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    notice the numbers 10 40 90. i hope you
  • 00:02:45
    get a little q in there there's one
  • 00:02:47
    there's four there's nine if we divide
  • 00:02:50
    each one by ten
  • 00:02:51
    now i am going to set them into
  • 00:02:54
    oscillation
  • 00:02:55
    and i am going to count say 20
  • 00:02:58
    vibrations of each one
  • 00:03:00
    here's the way i would do it
  • 00:03:02
    i would set this one into vibration
  • 00:03:05
    starting with zero and here is a clock
  • 00:03:08
    which i would start and stop
  • 00:03:10
    and i will just run one a moment to give
  • 00:03:13
    you the cue
  • 00:03:14
    i would remind you parenthetically
  • 00:03:17
    i am not really giving you a lecture in
  • 00:03:20
    physics i am merely pointing out some
  • 00:03:22
    things that you yourself can do with
  • 00:03:24
    your teacher and even at home
  • 00:03:26
    and my purpose is singularly this to
  • 00:03:30
    invite your interest and stir your
  • 00:03:32
    enthusiasm and curiosity
  • 00:03:35
    and point up at the same time the beauty
  • 00:03:37
    and drama in these things so i would
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    start this
  • 00:03:41
    zero
  • 00:03:43
    zero one two three four and i would
  • 00:03:47
    count 20 oscillations
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    and what would i get probably in this
  • 00:03:53
    laboratory i would get about 13 seconds
  • 00:03:57
    now where you are say colorado on a high
  • 00:04:00
    mountain you would get a different time
  • 00:04:02
    and if you were in australia you would
  • 00:04:03
    get a different time because little g is
  • 00:04:06
    different from place to place on the
  • 00:04:07
    earth so i count 13 or 20 oscillations
  • 00:04:12
    and i get 13 seconds now i do it all
  • 00:04:15
    over with this one
  • 00:04:19
    and i count 20 oscillations and what
  • 00:04:21
    would i get i would get about 26 seconds
  • 00:04:27
    you're thinking of something
  • 00:04:28
    then i would do it with the 90
  • 00:04:30
    centimeter one zero
  • 00:04:32
    one
  • 00:04:33
    oh i should have stopped this thing
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    because i'm not really clocking
  • 00:04:39
    yeah what's going on here
  • 00:04:42
    huh
  • 00:04:44
    well there's something wrong with the
  • 00:04:46
    notice now this is this is the hazard we
  • 00:04:49
    run when we depend on mechanical things
  • 00:04:51
    so i'm going to put the clock out of my
  • 00:04:53
    sight what would i do coming back here
  • 00:04:56
    somebody says isn't the professor having
  • 00:04:58
    a wonderful time with things going wrong
  • 00:05:00
    sure they are when you deal with nature
  • 00:05:03
    you must make her requirements
  • 00:05:04
    absolutely perfect or she will not do
  • 00:05:06
    what you want done so i
  • 00:05:08
    count
  • 00:05:10
    i count 20 oscillations and lo and
  • 00:05:13
    behold i get 39 seconds
  • 00:05:17
    look
  • 00:05:18
    10 40 90 they are in the ratio of one to
  • 00:05:22
    four to nine
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    13 26 39 they are in the ratio of one to
  • 00:05:29
    two to three
  • 00:05:31
    and now a marvelous thing is encountered
  • 00:05:34
    one is the square root of one
  • 00:05:36
    two is the square root of 4 and 3 is the
  • 00:05:39
    square root of 9. and that's what that
  • 00:05:42
    so-called formula says that the period
  • 00:05:46
    is proportional to the square root of
  • 00:05:48
    the length and this other stuff comes
  • 00:05:50
    out of some mathematical gymnastic so we
  • 00:05:53
    have learned a wonderful thing about
  • 00:05:54
    pendulums by examining the motion of
  • 00:05:57
    three of them
  • 00:05:59
    now strangely enough
  • 00:06:02
    when you make the exploration or develop
  • 00:06:05
    the formula mathematically notice notice
  • 00:06:09
    that it does not contain
  • 00:06:12
    it says nothing about
  • 00:06:14
    how big is the pendulum bob how massive
  • 00:06:17
    is it
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    all the period depends upon is its
  • 00:06:20
    length and you can show this by such an
  • 00:06:24
    adventure as i have here
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    here i have some
  • 00:06:28
    pendulum which are of identical length
  • 00:06:30
    let us say
  • 00:06:32
    here is one made of brass there is one
  • 00:06:34
    of aluminum there is one of cork and we
  • 00:06:37
    would find that if their lengths are
  • 00:06:39
    identical then their periods are
  • 00:06:41
    identical and that's a wonderful thing
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    because wouldn't you think that the
  • 00:06:44
    heavier the bob
  • 00:06:47
    that
  • 00:06:47
    a different period would result
  • 00:06:50
    no it does not
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    so we deal with simple pendulum
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    pendulums a bob on the end of a string
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    now anything can be a simple pendulum
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    anything let me illustrate supposing i
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    took you took me here there i am and you
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    put a hook there in me and you you you
  • 00:07:10
    oscillated me
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    i am a pendulum but not a simple one i
  • 00:07:15
    am a complex one
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    or more more more uh better physics
  • 00:07:21
    language i am a physical pendulum
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    a physical pendulum
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    and what is a physical pendulum
  • 00:07:29
    anything that is not a simple pendulum
  • 00:07:32
    illustration
  • 00:07:33
    here is a metal rod
  • 00:07:36
    now if i put this on a support
  • 00:07:39
    at one end and i oscillate it
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    in a vertical plane it has a certain
  • 00:07:45
    period
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    it has a certain period
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    now remember if it has a certain period
  • 00:07:54
    about an axis through one end
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    it has a certain time for its
  • 00:07:58
    oscillation
  • 00:08:00
    must there not be a simple pendulum
  • 00:08:03
    which has the same period yes indeed a
  • 00:08:06
    simple pendulum
  • 00:08:08
    has the same period if it is so long now
  • 00:08:12
    how long is the simple pendulum that has
  • 00:08:16
    the same period as this rod
  • 00:08:19
    wonderful
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    if this rod is length l
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    the equivalent simple pendulum is
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    two-thirds of l
  • 00:08:28
    so if i had a simple pendulum hanging
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    here
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    in this fashion
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    which was two-thirds as long as the rod
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    and i put them into oscillation together
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    they would stay in phase meaning that
  • 00:08:43
    they would keep step instead
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    so we speak of the equivalent simple
  • 00:08:47
    pendulum as a simple pendulum which has
  • 00:08:50
    the same period as the physical pendulum
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    now this is a rod
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    and this has a marvelous property
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    if i turn it around
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    and put
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    it would be looking like this of course
  • 00:09:10
    now
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    here's the two-thirds mark
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    here is a remarkable thing absolutely
  • 00:09:16
    enchanting
  • 00:09:18
    this rod has the same
  • 00:09:20
    period for this axis
  • 00:09:24
    as for this one
  • 00:09:26
    the same period
  • 00:09:28
    and hardly anybody ever believes that
  • 00:09:30
    but it is true as you can get discover
  • 00:09:32
    by doing the experiment yourself
  • 00:09:35
    so now we explored a physical pendulum
  • 00:09:38
    which is a
  • 00:09:40
    rod
  • 00:09:42
    how about a hoop
  • 00:09:44
    here is a hoop
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    i have here a hoop
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    and i could support it by an axis about
  • 00:09:50
    one edge
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    and i could let it oscillate in a
  • 00:09:53
    vertical plane
  • 00:09:55
    i could let it oscillate in a vertical
  • 00:09:57
    plane
  • 00:09:58
    does it not have a certain period of
  • 00:10:00
    course
  • 00:10:01
    it therefore has an equivalent simple
  • 00:10:03
    pendulum
  • 00:10:04
    and how long is the equivalent simple
  • 00:10:06
    pendulum it is a marvelous thing to
  • 00:10:09
    discover but the equivalent simple
  • 00:10:11
    pendulum has a length equal to the
  • 00:10:13
    diameter of the hoop so if you put a
  • 00:10:16
    simple pendulum up here
  • 00:10:18
    which has this diameter this length
  • 00:10:21
    they would keep in phase
  • 00:10:23
    finally
  • 00:10:24
    how about a disc
  • 00:10:26
    oh a disc
  • 00:10:28
    here is a uniform circular plate and
  • 00:10:31
    could i not support it on an axis in the
  • 00:10:34
    manner as before
  • 00:10:36
    and let it swing in its own plane
  • 00:10:40
    supposing well i better draw a new
  • 00:10:42
    picture
  • 00:10:43
    we are talking about a circular disk
  • 00:10:46
    a uniform circular plate and i
  • 00:10:49
    oscillated about one edge
  • 00:10:52
    what is the equivalent simple pendulum
  • 00:10:54
    strangely enough the equivalent simple
  • 00:10:57
    pendulum is three quarters of the
  • 00:10:59
    diameter a matter which you can explore
  • 00:11:02
    by making some of these things yourself
  • 00:11:05
    now in addition to simple pendulum
  • 00:11:09
    that's a bob on a string
  • 00:11:13
    and physical pendulum which are bodies
  • 00:11:16
    of any shape like rods and discs and
  • 00:11:18
    hoops and the like there are many other
  • 00:11:21
    oscillating
  • 00:11:24
    devices consider for example a spring
  • 00:11:28
    here is a beautiful spring
  • 00:11:31
    nicely wound
  • 00:11:33
    uh so long when unloaded
  • 00:11:36
    i put a load on it
  • 00:11:38
    and i call your attention to the
  • 00:11:40
    beautiful motion which it executes
  • 00:11:44
    that has a certain period in terms of
  • 00:11:46
    the property of the spring now supposing
  • 00:11:49
    i put a heavier load on it a heavier
  • 00:11:51
    load
  • 00:11:53
    watch it
  • 00:11:56
    and i say that's a beautiful motion to
  • 00:11:59
    experience to witness
  • 00:12:01
    and now
  • 00:12:04
    the question a beautiful question here
  • 00:12:06
    is a spring of certain length and so
  • 00:12:08
    stiff here is an identical spring
  • 00:12:12
    which is only half as long
  • 00:12:14
    the question which arises is this
  • 00:12:17
    if the motion of this pendulum is so
  • 00:12:19
    much with a certain load
  • 00:12:21
    what would be the motion of half the the
  • 00:12:24
    spring with the same load
  • 00:12:27
    and it is a wonderful thing to discover
  • 00:12:29
    that it has
  • 00:12:31
    well i'm not going to say i'll leave it
  • 00:12:33
    as something for you to explore because
  • 00:12:35
    i think it has much more virtue to leave
  • 00:12:38
    some questions unanswered for you to
  • 00:12:40
    explore than to just give you the answer
  • 00:12:42
    so i suggest you get a spring indeed two
  • 00:12:45
    springs
  • 00:12:46
    and explore their motions when one is
  • 00:12:49
    twice as long as the other and indeed
  • 00:12:51
    you will notice that i have couple
  • 00:12:53
    springs in different ways
  • 00:12:56
    suggesting that their periodic motions
  • 00:12:59
    depend upon how they are coupled
  • 00:13:02
    regarding coupled pendulum look here
  • 00:13:05
    here i have a pendulum a heavy lead bob
  • 00:13:08
    on the end of a rod which is pretty
  • 00:13:11
    nearly a simple pendulum and here is
  • 00:13:13
    another one
  • 00:13:14
    and i have as we say couple them
  • 00:13:17
    by putting a spring between them
  • 00:13:20
    and watch the marvelous behavior i'm
  • 00:13:22
    going to start one
  • 00:13:25
    i'm going to start one
  • 00:13:27
    and let it swing
  • 00:13:29
    and we will see this enchantment
  • 00:13:31
    its motion will soon die out
  • 00:13:34
    and the other one will take up the
  • 00:13:36
    motion then that one will die out and
  • 00:13:38
    the other one will take up the motion
  • 00:13:40
    and this coupling of pendulums has much
  • 00:13:45
    to do with electric circuits about which
  • 00:13:48
    we shall talk another time but watch it
  • 00:13:52
    this one is going
  • 00:13:55
    that one is going more watch this one
  • 00:13:57
    stop
  • 00:14:00
    that has practically stopped
  • 00:14:02
    now this one is going to stop
  • 00:14:07
    it has practically stopped and this one
  • 00:14:09
    is going and so we speak of coupled
  • 00:14:11
    harmonic oscillators here is another
  • 00:14:14
    pair here are two weights on very
  • 00:14:17
    flexible rods if i start one
  • 00:14:21
    we will discover a wonderful thing
  • 00:14:24
    that the motion of the first one
  • 00:14:27
    is arrested and the energy is taken up
  • 00:14:29
    by the second one
  • 00:14:31
    so
  • 00:14:32
    i urge you to explore the behavior of
  • 00:14:35
    oscillating bodies and i thank you for
  • 00:14:37
    your attention
  • 00:14:40
    [Music]
  • 00:14:48
    you
タグ
  • pendulum
  • oscillation
  • physics
  • Julius Sumner Miller
  • period
  • simple pendulum
  • physical pendulum
  • springs
  • coupled pendulums
  • experimentation