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(#79) Calculus and Dynamics for Mechanical Design Trust · Mechanical Engineering Made Simple

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  Welcome back, design junkies, to mechanical engineering made
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  simple. We are we're diving deep today
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  and our mission is a big one. We really want to unlock the
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  foundational mathematics, you know, the core concepts of
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  mechanics that that really underpin all successful
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  mechanical design. Yeah, and this is for you, the
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  listener. We're aiming to show why
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  engineers trust these specific tools, these calculations from,
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  I mean from abstract algebra all the way to real world dynamics.
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  Trust is exactly the right word, because if you look at modern
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  engineering, it's all computational fluid dynamics
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  finite element analysis. Right, this incredibly complex
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  software. Exactly.
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  But that software is only as good, only as reliable as the
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  math it's built on. And our materials today show
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  this, this just incredible shift in the tools over time.
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  It's a huge contrast. I mean, you start with it now
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  seems like, well, obsolete technology.
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  Oh. For sure, like the old school
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  Kufel and Esser slide rule, a marvel for its time, but it's
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  all based on a manual logarithmic process.
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  Very easy to make a mistake. And you put that K&E slide rule,
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  which took so much focus just to avoid a gross error, next to a
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  modern programmable calculator, something like Texas Instruments
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  TI-85 or an HP 48G. It's not even a fair fight.
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  It's not just a leap in speed, it's a total revolution in
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  precision. So what was the point?
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  Why make that jump? Well, the whole purpose of
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  embracing that digital power was to move beyond those error prone
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  procedures. Before these tools, every single
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  calculation was this this cumbersome, tedious exercise.
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  The power of modern handheld calculation is that it
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  simplifies is these hugely complex steps while at the same
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  time cranking up our precision. And that, in turn, is what
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  allows us to trust the final design.
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  OK, so let's unpack that foundation.
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  To trust, we have to look at the invisible engine.
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  You know, the thing driving all those equations we see in
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  engineering handbooks. The ones for beams, shafts,
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  fluid flow, all of them. Yeah, those simple looking
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  rules, they were derived by mathematicians using calculus,
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  using other advanced pro futures.
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  Absolutely. At the very bedrock of modern
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  mechanics is differential calculus, and specifically the
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  derivative. For those of us who maybe
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  haven't touched a math textbook in a while, what's the core
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  insight of the derivative for an engineer?
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  Simply put, the derivative is the rate of change of function.
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  That's it. It is the most powerful tool we
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  have for analysis for optimization.
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  For a mechanical engineer, it really boils down to two key
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  interpretations. First, it tells you the slope of
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  a continuous curve at one specific point.
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  This is how we understand how quickly a design dimension is
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  changing. So if you were trying to find,
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  say say, the weakest point on a curve or the most efficient
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  point, the derivative shows you where that slope is at a maximum
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  or minimum. The second key interpretation is
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  velocity and acceleration. If distance, which we can call S
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  is a function of time T, then the first derivative of that
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  function S of T gives you the velocity.
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  And if you take the derivative. Again, the 2nd derivative,
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  right? The derivative of the velocity
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  function gives you acceleration. That makes the physics
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  connection really clear. Our notes actually have the
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  equation for a body falling to earth if you ignore air
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  resistance S over time, T = K * t ^2, and here K is 16 feet per
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  second squared. Right.
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  And now if you take the second derivative of that function, you
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  land squarely on the constant for acceleration due to gravity.
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  32 feet per second squared. 32 or the value engineers rely on
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  for standard practice, which is 32.16 feet per second squared.
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  That little difference that .16. That's the difference between
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  textbook physics problem and a real world engineering
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  calculation. So the derivative is all about
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  the rate of change, then the integral, the anti derivative,
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  it has to be about accumulation. Right it is.
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  Integrals are the language of accumulation.
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  We use definite integrals which are defined by the fundamental
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  theorem of calculus. The integral of a function F of
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  X from point A to point B is just the anti derivative at B
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  minus the anti derivative at a. That sounds very abstract.
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  Where is an engineer actually using that on a design?
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  Oh, everywhere. Especially when you're trying to
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  figure out physical quantities of weirdly shaped objects.
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  Think about calculating the area under a parabolic curve.
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  For instance, let's say the equation is X = y ^2 + 1.
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  If you need the exact area projected to the Y axis between,
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  say y = -2 and y = 2, integration is really the only
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  practical way to do it. And without calculus, you'd be
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  what, drawing a million tiny rectangles and adding them up by
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  hand? Which is basically what the
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  integral does for you, just instantly.
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  And that integration gives us a precise answer. 9.333 square
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  units use the same idea for volumes, centroids and controid.
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  This is a big one we'll get to later.
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  Yeah, moments of inertia. OK, moving past calculus for a
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  second, our notes also point to some specialized algebra that's
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  still fundamental. I'm curious about complex
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  numbers. They always felt a little like a
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  mathematician's trick with that imaginary unit IR.
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  It feels like a trick, but it's a necessary one.
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  Especially in electromechanical systems and vibration analysis.
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  We have to rely on that imaginary unit I where I ^2 = -1
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  real numbers. They only tell you magnitude.
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  But complex numbers do more. They convey phase and magnitude
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  at the same time. When you're plotting the
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  behavior of a spinning shaft or analyzing A vibration, you need
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  to map not just how big it is, but it's direction.
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  So how does that look in a design?
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  That's where we use the polar form.
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  A plus by equals R times the quantity cosine Theta plus I
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  sine Theta. That R gives you the magnitude
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  and Theta gives you the direction.
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  The phase shift it's it's essential for making sure any
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  rotating mechanism is stable. And one last quick but important
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  bit of algebra. Logarithms.
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  Even with all our computers, they're still really efficient
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  for solving compound rate problems.
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  Oh, absolutely. The notes have a great example.
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  Say a company's employment is growing at 15% per year.
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  How long until the number of employees triples?
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  You can guess or. Or you use logarithms.
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  The time end is just log 3 divided by log of 1.15 and it
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  comes out to 7.86 years. It's just a very elegant way to
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  handle growth and decay. So now we jump from the abstract
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  math to the science of applying it.
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  Engineering mechanics. This is where the numbers really
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  meet the machine and it's split into statics and dynamics.
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  Right. Statics is the study of bodies
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  in equilibrium. That means they're either
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  totally at rest or moving at a constant uniform motion, no
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  acceleration. And dynamics then is the
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  opposite. It is.
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  Dynamics is the study of bodies that are not in equilibrium, so
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  they're speeding up, slowing down, changing direction.
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  But both of them rely on the same basic ideas of force,
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  specifically moment in torque. They do.
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  A moment is just a force multiplied by its moment arm.
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  That's the perpendicular distance from the force to the
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  pivot point. Torque is, well, it's the
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  technical term for a forces tendency to cause rotation.
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  They seem very similar. They are.
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  The context is usually what separates them.
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  We tend to say torque when we're talking about rotational work,
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  like what an engine produces. OK, before we get into the
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  calculations, we have to address something that causes huge
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  confusion, the difference between the US customary system
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  and the SI system, specifically with mass and force.
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  Oh, this is this is more than just a naming issue.
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  It is a philosophical difference that has led to actual
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  catastrophic failures. the US system derives mass from force.
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  The standard unit of mass is the slug.
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  The slug. Not something you hear everyday.
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  No, and it's defined as weight W divided by gravity G, which is
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  32.2 feet per second squared. The real problem though, is that
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  the term pound gets used for everything.
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  Pound force and pound mass. Exactly.
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  And they are not the same thing. If you mix those up in a
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  critical calculation, you can be off by a factor of 32.
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  It's it's a huge source of error.
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  And the SI system is much. Cleaner.
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  It is so much cleaner, it's elegant.
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  The SI system derives force from mass.
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  The kilogram is mass, period. The Newton is force, period. 150
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  kilogram mass exerts a force of m * G Newton.
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  So 150 * 9.81 it's 14171.5 Newtons.
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  No confusion. No confusion.
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  Keeping them distinct is absolutely non negotiable for
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  safety. So when we're analyzing a static
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  system, say a simple supported beam, we're looking for
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  equilibrium. What are the rules that
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  guarantee it's stable? There are two main conditions.
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  First, the resultant of all forces acting on the body has to
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  be 0. No net push, no net pull.
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  Second, the algebraic sum of all the moments of all forces about
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  any point you choose also has to be 0.
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  That guarantees it won't rotate. And to check those conditions,
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  engineers are always breaking down forces that are at an angle
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  right into simpler components. That's called force resolution.
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  You break the force down along the X&Y axis, so the force in
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  the X direction F sub X is just the total force F times the
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  cosine of the angle Phi, and F sub Y is F times the sign of
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  Phi. Just turns a tricky geometry
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  problem into a simple right triangle problem.
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  Let's use the classic example from our notes.
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  A supported beam with a few different loads pushing down on
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  it. 2 pounds, 3 lbs, and 25 lbs. And there's an upward support
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  force F that we need to find. Yep, and the fastest way to
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  solve this is to use that second condition of equilibrium.
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  We take the algebraic sum of the moments about that fixed pivot
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  point point A and we set that sum to 0.
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  So you calculate the moment from the 2 LB load, the 3 LB load,
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  the 25 LB load, and balance them against the moment from our
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  unknown force F. Using the specific distances
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  from the diagram, yes, And when you work out the simple algebra.
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  The support force F has to be exactly 221.14 lbs to keep the
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  whole thing in equilibrium. That moment calculations what
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  ensures it doesn't pivot. It's a perfect, simple
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  illustration of how we use that math to guarantee physical
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  stability. And now we enter the world of
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  dynamics, where things are moving, accelerating.
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  And this is where the idea of inertia becomes so important.
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  We need to talk about moments of inertia.
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  Right, which is just the term for a body's resistance to
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  rotational change. And they come in two main
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  flavors, both derived from the product of little elementary
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  bits of mass or area and the square of their distance from an
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  axis. OK, first up is the polar moment
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  of inertia of areas I sub PA. This one is for plane areas.
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  Think cross sections of beams or columns.
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  It's about how resistant the shape is to twisting you.
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  Just sum the moments about The xx and YY axis.
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  But for things in motion, the one we really care about is the
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  polar moment of inertia of masses.
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  I sub PM. That's the one I sub PM is used
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  in all dynamics equations involving rotation.
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  It's what tells you how much effort it takes to start
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  spinning a flywheel or how hard it is to stop it.
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  The units in the SI are kilogram meter squared.
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  A lot of times in a real machine, a part isn't spinning
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  right around its own center of gravity.
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  That's where the parallel axis theorem is used, right?
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  It is absolutely vital the theorem, or what we call the
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  transfer of axes, lets you accurately shift your
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  calculation. You can move from the
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  component's own center of gravity axis, call it Z prime, Z
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  prime, to the main axis. It's actually rotating around
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  ZZ. And the formula for that is I
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  sub PM equals I prime sub PM plus s ^2 * m.
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  Right, and that S is the distance between the two axis
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  and M is the mass. You have to account for that
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  offset. If you don't, your calculation
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  for rotational resistance will be completely wrong.
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  OK. And this is where it all comes
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  together. This is where all this abstract
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  math gets connected to a real complex industrial product.
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  Our case study. The design analysis of a high
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  power electrical vacuum interrupter switch.
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  This is the aha moment. This example shows exactly why
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  we do all this, and it embodies a critical design philosophy.
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  Function have to be the primary goal.
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  It comes before costing, before tooling.
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  Because if you prioritize cost first.
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  You risk product failure and on a high power switch like this,
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  failure is not an option. So the designer focuses the
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  analysis only on the most critical elements where a
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  failure could happen. So we're talking about
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  calculating the energy and the power springs, the forces on the
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  crank arm, the torque on the main power shaft.
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  Yes, and even the current carrying ability of the copper
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  blades. But let's focus on the energy
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  comparison calculation, because that's the ultimate validation.
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  OK, what's the comparison? This mechanism works by
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  converting the potential energy stored in some big power springs
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  into the kinetic energy of rotation.
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  In six heavy copper blades, we just compare the input energy to
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  the output energy. So start with the input the
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  potential energy E sub P. Right, we use the spring
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  potential energy equation R sub S * s ^2 / 2.
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  We calculate that for the three main springs in the mechanism,
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  we sum them up and the total input energy is 11,175 inch
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  pounds. That's the fuel.
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  Now for the output, the kinetic energy of rotation E sub KR.
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  And for that we use the inertia concept we just discussed.
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  I sub PM times Omega squared divided by two.
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  We take the mass moment of inertia for those six blades and
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  their angular velocity, which was calculated to be 27.1
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  radians per second and. The total output kinetic energy
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  is. 1128 inch pounds. So 11,175 inch pounds went in
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  and 11128 inch pounds came out. Exactly.
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  The math validates the physical reality.
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  There's a reduction in energy of exactly 4%.
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  And that's not an error in the math.
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  No. That small loss is realistically
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  and predictably attributed to friction in the mechanism.
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  The systematic calculation rooted in calculus and dynamics
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  proves the mechanism is reliable and efficient before they ever
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  build the expensive final version.
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  That, for me, is the absolute core power of engineering.
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  You're taking these complex invisible forces and making a 4%
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  friction loss a predictable mathematical result.
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  It's not a guess. You have to fix with expensive
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  trial and error. We've seen how calculus gives us
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  the principles, how statics ensures stability, and how
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  dynamics with moments of inertia predicts precise motion.
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  It all feeds back to that idea of trust in the numbers.
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  So what does all this predictability suggest for the
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  world around us? Well, our source materials
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  mention Newton's laws of motion, specifically that a particles
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  acceleration is proportional to the resultant force acting on
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  it. And when we look at that complex
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  switch or really any machine, we take that resultant force for
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  granted. It's not one push or one pull.
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  It's the sum of everything. It's the algebraic sum of
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  hundreds of forces and moments that determines if a system is
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  stable, how fast it goes, if it's safe.
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  So I'd ask you to think about this.
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  How often do we rely on these unseen mathematical checks?
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  What other essential geometric or algebraic concepts like like
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  trigonometries, mall wide equation for checking triangles
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  or using determinants to solve force systems, What other checks
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  are providing this critical unseen safety net in the world
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  we all live in? Fascinating thought that
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  predictability is what lets us trust the buildings we're in and
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  the machines we use every day. If you want to dive deeper into
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  the math and the mechanics we covered, definitely check out
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  the bibliographical references listed in our sources for some
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  further study.