The Equations of Motion part 1: Acceleration

00:15:42
https://www.youtube.com/watch?v=JgfY6RGiqzU

Zusammenfassung

TLDRThis talk explains the equations of motion in the ocean with a focus on acceleration, influenced by various forces. Starting with Newton's second law (F=ma), it illustrates how acceleration is handled in fluid dynamics using density. The discussion includes acceleration components in three dimensions (U, V, W), and emphasizes the importance of both local acceleration and advection terms. The session sets the foundation for understanding how external forces influence ocean motion, leading to further exploration of specific forces in upcoming discussions.

Mitbringsel

  • 🔍 Newton's second law: F = ma governs motion.
  • 🌊 Acceleration in fluids is density-based.
  • 📏 Velocity has three components: U, V, W.
  • 🔄 Local acceleration is time-dependent change.
  • 🌐 Field acceleration looks at spatial differences.
  • 📈 Advection terms arise from changing spatial velocity.
  • ⚖️ Use density instead of total mass in fluids.
  • 🌪️ Coriolis effect alters motion direction.
  • ⬆️ Pressure gradient force drives flow in oceans.
  • 🔄 Acceleration cumulatively affects ocean movement.

Zeitleiste

  • 00:00:00 - 00:05:00

    The talk introduces the equations of motion, focusing on acceleration in the ocean and the forces affecting it. It emphasizes Newton's second law, stating that acceleration is directly proportional to the net force acting on an object and inversely proportional to its mass. In the oceanic context, mass is discussed in terms of density and force per unit volume, leading to the expression of acceleration as dependent on the net forces acting on the water.

  • 00:05:00 - 00:10:00

    Continuing from the previous discussion, the presenter elaborates on the definition of acceleration as the change in velocity over time. The video describes the three components of the velocity vector in the ocean, namely u (east-west), v (north-south), and w (vertical). It explains that acceleration in each direction can be represented as the time derivative of these velocity components, emphasizing the importance of total derivatives due to the dependence of velocity on multiple spatial variables in addition to time.

  • 00:10:00 - 00:15:42

    Finally, the presenter differentiates between local acceleration, resulting from temporal changes in velocity, and field acceleration or advection, which arises from spatial changes in velocity across the ocean. The discussion concludes by indicating that upcoming talks will cover other forces like gravity and pressure gradient force that interact with these acceleration components in ocean dynamics.

Mind Map

Video-Fragen und Antworten

  • What are the main forces affecting ocean motion?

    The main forces include pressure gradient force, gravity, Coriolis effect, and friction.

  • What does F = ma represent?

    F = ma is Newton's second law, stating that force equals mass times acceleration.

  • How is acceleration calculated in fluids?

    Acceleration is calculated using density and force per unit volume in fluid dynamics.

  • What are the components of velocity in the ocean?

    The components of velocity are U (east-west), V (north-south), and W (up-down).

  • What is local acceleration?

    Local acceleration is the change in velocity with respect to time for a specific location.

  • What are advection terms?

    Advection terms are changes in velocity due to spatial variations in flow within the fluid.

  • Why can't we use mass for fluids directly?

    For fluids, we use mass per unit volume (density) instead of total mass.

  • How does pressure gradient force affect ocean motion?

    Pressure gradient force drives movement from areas of high pressure to low pressure.

  • What role does the Coriolis effect play in ocean dynamics?

    The Coriolis effect influences the direction of fluid motion due to Earth's rotation.

  • How do you differentiate between local and field acceleration?

    Local acceleration is time-dependent change at a point, while field acceleration considers spatial variations.

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Untertitel
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Automatisches Blättern:
  • 00:00:02
    hello welcome back in this talk we are
  • 00:00:08
    going to talk about the equations of
  • 00:00:11
    motion these are the equations and that
  • 00:00:15
    governed all the motion in the ocean
  • 00:00:20
    in this talk we're going to focus on
  • 00:00:23
    acceleration and the acceleration terms
  • 00:00:26
    that affect the motion in the ocean then
  • 00:00:30
    in the next talk we're going to see how
  • 00:00:31
    the acceleration is affected by the
  • 00:00:35
    pressure gradient force gravity Coriolis
  • 00:00:39
    friction and other forces talking about
  • 00:00:44
    forces let's think about forces again
  • 00:00:48
    for a second so what do forces do well
  • 00:00:53
    if we think back to lab one we saw that
  • 00:00:57
    force is equal to mass times
  • 00:00:58
    acceleration F equals MA that is
  • 00:01:02
    Newton's second law and we can put that
  • 00:01:05
    into words and we see that a net force
  • 00:01:08
    causes an object to accelerate with an
  • 00:01:12
    acceleration directly proportional to
  • 00:01:14
    the size of the force so looking over
  • 00:01:17
    here at this diagram we see that the
  • 00:01:19
    acceleration here on the y axis and the
  • 00:01:22
    force on the x axis if we increase our
  • 00:01:26
    force we increase our acceleration
  • 00:01:30
    proportionately so if we increase the
  • 00:01:33
    force the acceleration increases
  • 00:01:35
    proportionally we have a linear
  • 00:01:37
    relationship between force and
  • 00:01:39
    acceleration so if you push harder on
  • 00:01:43
    something it will accelerate more this
  • 00:01:48
    is Newton's second law F equals MA so
  • 00:01:54
    now for a particle of mass M if we
  • 00:01:56
    divide through by m right here we end up
  • 00:02:00
    with the acceleration equals 1 over m
  • 00:02:03
    times the sum of those forces F net all
  • 00:02:07
    right we're going to use this expression
  • 00:02:09
    right here to develop the equations of
  • 00:02:12
    motion that govern the ocean all right
  • 00:02:19
    so we're
  • 00:02:19
    starting with the idea that acceleration
  • 00:02:21
    equals force over mass Newton's second
  • 00:02:24
    law and that's great when we're talking
  • 00:02:27
    about a baseball or a box when we're
  • 00:02:31
    looking at something simple we're doing
  • 00:02:33
    it our intro physics classes however
  • 00:02:35
    we're talking about the ocean and the
  • 00:02:38
    ocean isn't a ball it's not a box it's a
  • 00:02:42
    fluid so we can't talk about its mass
  • 00:02:45
    instead we talk about its mass per unit
  • 00:02:48
    volume
  • 00:02:50
    similarly we talk about the force per
  • 00:02:53
    unit volume so the mass per unit volume
  • 00:02:56
    we know is the density where the density
  • 00:03:01
    is the mass over the volume and if we
  • 00:03:04
    rewrite this expression using instead of
  • 00:03:07
    mass using the mass per volume or the
  • 00:03:10
    density and instead of force using the
  • 00:03:13
    force per volume we get that our
  • 00:03:16
    acceleration equals 1 over the density
  • 00:03:18
    times the force per volume and just to
  • 00:03:22
    check to make sure that this doesn't
  • 00:03:23
    affect satisfy Newton's second law we
  • 00:03:27
    can put bring this expression down here
  • 00:03:29
    and we're going to now replace the
  • 00:03:35
    density right here I'm going to replace
  • 00:03:37
    this density by a mass over the volume
  • 00:03:41
    and it's 1 over that so instead of
  • 00:03:45
    writing mass over volume if I put 1 over
  • 00:03:48
    that I end up with the volume over the
  • 00:03:50
    mass multiplying that by the force over
  • 00:03:55
    the volume and I realize I can cancel my
  • 00:03:59
    volumes and so what I'm left with right
  • 00:04:03
    here is equal to 1 over the mass times
  • 00:04:07
    the force which is exactly what we had
  • 00:04:10
    to begin with Newton's second law
  • 00:04:14
    alright so using our expression let's
  • 00:04:18
    get started alright so we have inwards
  • 00:04:21
    here this acceleration right here is
  • 00:04:25
    equal to 1 over the density times the
  • 00:04:29
    sum of the forces and those forces we're
  • 00:04:31
    familiar with those now
  • 00:04:32
    gravity and pressure gradient force in
  • 00:04:35
    Coriolis and friction and any other
  • 00:04:38
    forces that are acting in the direction
  • 00:04:40
    that we are interested in so let's focus
  • 00:04:43
    first on the acceleration all right we
  • 00:04:48
    remember back to lab one again that the
  • 00:04:51
    acceleration is equal to the change in
  • 00:04:53
    the velocity with time so it's the
  • 00:04:58
    change in the velocity with time and we
  • 00:05:01
    can write that as the acceleration is
  • 00:05:05
    equal to the change in the velocity over
  • 00:05:17
    the change in time and we remember that
  • 00:05:28
    if we allow delta T to go to zero this
  • 00:05:33
    becomes
  • 00:05:42
    dv/dt we can also remember that the
  • 00:05:48
    velocity itself is the change in the
  • 00:05:51
    position over time so if we think about
  • 00:05:57
    this velocity here as the change in the
  • 00:06:01
    position with time we can get that the
  • 00:06:05
    velocity equals the change in the
  • 00:06:09
    position which I'll go ahead and do as a
  • 00:06:11
    Y so that'll be dy DT alright let's do a
  • 00:06:18
    little review reminding ourselves about
  • 00:06:20
    this velocity and how we talk about the
  • 00:06:23
    velocity vector in the ocean so we
  • 00:06:27
    remember that the velocity vector in the
  • 00:06:29
    ocean is divided into three different
  • 00:06:32
    dimensions and the first direction that
  • 00:06:34
    we have is our X Direction or east-west
  • 00:06:36
    direction our zonal direction and for
  • 00:06:39
    that we talk about the u velocity that's
  • 00:06:48
    the U velocities for the X component of
  • 00:06:51
    our velocity now in the north-south or
  • 00:06:56
    the Y direction that's our Meridian ol
  • 00:06:58
    direction and what's important in this
  • 00:07:00
    case what we call this component of our
  • 00:07:03
    velocity is V so we use V in the
  • 00:07:05
    north-south direction and in the
  • 00:07:09
    vertical direction the up/down direction
  • 00:07:11
    the Z direction we use W so the three
  • 00:07:19
    components of our velocity vector u V
  • 00:07:23
    and W now if we look at this diagram
  • 00:07:28
    right here the other thing that we see
  • 00:07:30
    is that the velocity itself as I
  • 00:07:35
    mentioned is the change in the position
  • 00:07:40
    with time so we can also say that u
  • 00:07:44
    equals DX DT
  • 00:07:52
    and say that me equals T Y DT and we can
  • 00:08:02
    say that W equals D Z DT all right let's
  • 00:08:16
    start thinking about acceleration so the
  • 00:08:21
    acceleration in the x-direction is going
  • 00:08:25
    to be d u dt the acceleration the
  • 00:08:28
    y-direction will be DV DT and the
  • 00:08:31
    acceleration in the Z direction will be
  • 00:08:34
    DW DT I've used capital D's here because
  • 00:08:39
    what we're going to need to do is take
  • 00:08:41
    the total derivative so let's look at
  • 00:08:45
    that and understand what's going on by
  • 00:08:47
    starting in the x-direction so in the
  • 00:08:50
    x-direction we have d u DG again I said
  • 00:08:55
    this is called the total derivative
  • 00:08:57
    because U is not just a function of time
  • 00:09:01
    you could be a function of many things
  • 00:09:03
    it could be a function of X it could be
  • 00:09:07
    a function of Y it could be a function
  • 00:09:09
    of Z and it could also be a function of
  • 00:09:12
    time so if we want to look at how at U
  • 00:09:24
    as a function of I want take the
  • 00:09:27
    derivative of this knowing that U is a
  • 00:09:29
    function of X Y Z and T then if we take
  • 00:09:42
    the total derivative of U with respect
  • 00:09:44
    to time what we need to do is take the
  • 00:09:47
    partial derivative of U with respect to
  • 00:09:51
    X first and then we can't just do that
  • 00:09:56
    because we need the derivative with
  • 00:09:57
    respect to time so we have to do the
  • 00:10:00
    chain rule and du DX DT
  • 00:10:07
    now we also need to take the different
  • 00:10:11
    partial derivative of U with respect to
  • 00:10:14
    Y and we also need to take the chain
  • 00:10:19
    rule so we do do you.why DT and we can
  • 00:10:27
    take the partial derivative of U with
  • 00:10:32
    respect to Z and again the chain rule
  • 00:10:37
    gives us DZ
  • 00:10:41
    tt and then we still now to take do you
  • 00:10:48
    tt and we can write the chain rule here
  • 00:10:51
    if we want to TT TT but we know that's
  • 00:10:56
    just one so what we get is du DT equals
  • 00:11:03
    D u DX DX DT Plus D u dy dy DT Plus D u
  • 00:11:09
    DZ DZ DT Plus D u DT now what we notice
  • 00:11:15
    is that as we just pointed out right
  • 00:11:21
    here we have DX DT well that we know is
  • 00:11:26
    just u so this part right here is just
  • 00:11:30
    du u DX times you alright well right
  • 00:11:37
    here we have two Y DT well that we just
  • 00:11:41
    saw is V so this is do you do you Y
  • 00:11:47
    times V and right here we have DZ DT and
  • 00:11:54
    we know that that is just W so this is
  • 00:11:58
    do you DZ times W and then right here we
  • 00:12:06
    just have do you DT so this is now the
  • 00:12:15
    total derivative that we get when we
  • 00:12:18
    bring all of these pieces together all
  • 00:12:21
    right so if we take control the
  • 00:12:25
    derivative of U with respect to T and we
  • 00:12:28
    get what we just had a second ago then
  • 00:12:32
    we can rewrite this and bring Beauty to
  • 00:12:37
    the front and just rearrange our terms
  • 00:12:39
    so we have du DT plus u 2 u DX plus V du
  • 00:12:46
    u dy plus w du u DZ and this is what we
  • 00:12:50
    call the total derivative so expanding
  • 00:12:56
    that out in all three dimensions we see
  • 00:13:01
    what we have 4x and then we can also
  • 00:13:05
    look how we can break this down the very
  • 00:13:10
    first term here Dutt that is what we
  • 00:13:14
    call our local acceleration that's if we
  • 00:13:17
    were just floating along with the water
  • 00:13:20
    parcel that's how it is changing in time
  • 00:13:25
    the other terms these terms here are our
  • 00:13:29
    field acceleration terms this is the
  • 00:13:34
    advection terms but what we mean here is
  • 00:13:38
    if we're looking at this as a field and
  • 00:13:41
    we're looking out at the ocean at
  • 00:13:43
    different points and we see that the
  • 00:13:45
    velocity might change in might be
  • 00:13:50
    changing spatially and so it is the
  • 00:13:52
    these are the infection terms or the
  • 00:13:54
    field acceleration terms looking at
  • 00:13:58
    these pictorially we see right here we
  • 00:14:05
    have our local acceleration which is
  • 00:14:07
    just the simple acceleration that we
  • 00:14:09
    talk about normally where we have our
  • 00:14:12
    change in our velocity with time and
  • 00:14:16
    that gives us an acceleration so in this
  • 00:14:19
    example if as we're going from point X
  • 00:14:23
    to point point x1 to point x2 to point X
  • 00:14:26
    3 we see that we're going there faster
  • 00:14:29
    from point x1 to point x2 then we are
  • 00:14:33
    from point x2 to point X 3 so v1 is
  • 00:14:36
    faster than v2 and so our acceleration
  • 00:14:40
    is in the opposite direction this is our
  • 00:14:43
    local acceleration this is d UDT if we
  • 00:14:48
    look at the infection terms what we see
  • 00:14:51
    in this case is that as we're changing
  • 00:14:54
    the field as we're moving to a different
  • 00:14:57
    point in the field that is when our
  • 00:15:02
    velocity is changing if we're changing
  • 00:15:05
    in terms of where we are in X we see
  • 00:15:09
    that the velocity changes based on where
  • 00:15:13
    we are in X so those are the advection
  • 00:15:17
    terms or the field acceleration terms
  • 00:15:21
    all right so that's that's all I have
  • 00:15:25
    for you for acceleration next we're
  • 00:15:28
    going to get into looking at the rest of
  • 00:15:31
    these terms we have gravity pressure
  • 00:15:35
    gradient force Coriolis for
  • 00:15:38
    and other forces
Tags
  • ocean dynamics
  • acceleration
  • Newton's law
  • fluid mechanics
  • density
  • velocity
  • forces
  • pressure gradient
  • Coriolis effect
  • advection