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    Maths derail from Mental Fitness

    Quote Originally Posted by BKR View Post
    The integral of power over time defines the work performed. Because this integral depends on the trajectory of the point of application of the force and torque, this calculation of work is said to be path dependent.
    Something about that wording - specifically "path dependent," - bothers me. Not that it's incorrect, if I understand what's being said, but that it's self-evident. If we're talking about a definite integral, then it's the sum of all work performed between two points in time, let's say t0 for start of work and t1 for end of work. The path is then defined by |Δt| (i.e. the passage of time) and augmented by any acceleration.

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    Quote Originally Posted by submessenger View Post
    Something about that wording - specifically "path dependent," - bothers me. Not that it's incorrect, if I understand what's being said, but that it's self-evident. If we're talking about a definite integral, then it's the sum of all work performed between two points in time, let's say t0 for start of work and t1 for end of work. The path is then defined by |Δt| (i.e. the passage of time) and augmented by any acceleration.
    Not quite.
    Path dependence is not a function of time but of direction.
    For example, in a baseball home run hit, the force applied from the bat to the ball along the trajectory the ball winds up traveling is part of the path, but, for example the force applied during a foul ball slice is much less no matter how hard the swing is because the angle of contact with the initiating force is steeper and results in less surface area of contact, so no matter how hard you swing if you hit the ball at the wrong angle you will always deliver less force.

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    Quote Originally Posted by AcerTempest View Post
    Not quite.
    Path dependence is not a function of time but of direction.
    For example, in a baseball home run hit, the force applied from the bat to the ball along the trajectory the ball winds up traveling is part of the path, but, for example the force applied during a foul ball slice is much less no matter how hard the swing is because the angle of contact with the initiating force is steeper and results in less surface area of contact, so no matter how hard you swing if you hit the ball at the wrong angle you will always deliver less force.
    But, direction is useless without time. It's merely the derivative at some point of work.

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    Quote Originally Posted by submessenger View Post
    Something about that wording - specifically "path dependent," - bothers me. Not that it's incorrect, if I understand what's being said, but that it's self-evident. If we're talking about a definite integral, then it's the sum of all work performed between two points in time, let's say t0 for start of work and t1 for end of work. The path is then defined by |Δt| (i.e. the passage of time) and augmented by any acceleration.
    My point is to help define "mental strength" to better understand how to "develop" it, in terms of the original analogy used by Phrost.

    One definition would be to get other people to do your thinking for you...
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    Quote Originally Posted by AcerTempest View Post
    Not quite.
    Path dependence is not a function of time but of direction.
    For example, in a baseball home run hit, the force applied from the bat to the ball along the trajectory the ball winds up traveling is part of the path, but, for example the force applied during a foul ball slice is much less no matter how hard the swing is because the angle of contact with the initiating force is steeper and results in less surface area of contact, so no matter how hard you swing if you hit the ball at the wrong angle you will always deliver less force.

    Thinking about this probably more than I should. In order for us to be on the same page, I'm talking about a discrete unit of work - either the swinging of the bat, or the environment acting on a post-collision ball in flight (or, perhaps, the ball's resistance to the environment while in flight).

    After the collision of ball and bat (which are two discrete systems of work: a) bat swings; b) ball flies) it is assumed that we're only talking about the ball in flight. So, the collision is t0 and the ball at rest is t1. Direction, in my scenario, describes the change in post-collision acceleration of the ball from moment to moment, until it comes to a rest. That entire function will always be "path dependent," which is why I thought it was unnecessary to mention it.

    In other words, work performed can always be expressed as force applied, and v/v, but force itself cannot determine what work is going to be done. Thus, "path dependent," is a redundancy in the description.

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    Quote Originally Posted by submessenger View Post
    Thinking about this probably more than I should. In order for us to be on the same page, I'm talking about a discrete unit of work - either the swinging of the bat, or the environment acting on a post-collision ball in flight (or, perhaps, the ball's resistance to the environment while in flight).

    After the collision of ball and bat (which are two discrete systems of work: a) bat swings; b) ball flies) it is assumed that we're only talking about the ball in flight. So, the collision is t0 and the ball at rest is t1. Direction, in my scenario, describes the change in post-collision acceleration of the ball from moment to moment, until it comes to a rest. That entire function will always be "path dependent," which is why I thought it was unnecessary to mention it.

    In other words, work performed can always be expressed as force applied, and v/v, but force itself cannot determine what work is going to be done. Thus, "path dependent," is a redundancy in the description.
    Kind of, but that doesn't quite work either. Because force can act in a non-path dependant way on an object already in motion, for example gravity acting on our flying ball is actually NOT path dependent on earth anyway because it acts equally on the object regardless of its position, direction or speed of travel. At least in regards to objects in proximity to the earth.

    And there are other environmental factors. Or an overwhelmingly LARGE force can act on an object in a way that is not path dependant. Or at least isn't for all practical purposes.

    For example, technically gravity doesn't work the way I have described it, but for the limited set of objects in close proximity to the earth, say within a few hundred miles of the surface, it does.

    Yes, the function is already always path dependent when plugging in normal forces acting through individual agency, but any agency that can act as a single force in multiple directions on the same object would then invalidate the functions descriptive use.
    If we state outright that its path dependence is a separate function of direction of force, we can then expand the equation to describe such forces by simply running it again for each direction. Clunky, but it can work.

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    BTW, this is WAY more physics than I was prepared for today. Thankfully my bottle of Glenlivet is still in my trunk so I'm good. :D

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    Quote Originally Posted by AcerTempest View Post
    Kind of, but that doesn't quite work either. Because force can act in a non-path dependant way on an object already in motion, for example gravity acting on our flying ball is actually NOT path dependent on earth anyway because it acts equally on the object regardless of its position, direction or speed of travel. At least in regards to objects in proximity to the earth.

    And there are other environmental factors. Or an overwhelmingly LARGE force can act on an object in a way that is not path dependant. Or at least isn't for all practical purposes.

    For example, technically gravity doesn't work the way I have described it, but for the limited set of objects in close proximity to the earth, say within a few hundred miles of the surface, it does.

    Yes, the function is already always path dependent when plugging in normal forces acting through individual agency, but any agency that can act as a single force in multiple directions on the same object would then invalidate the functions descriptive use.
    If we state outright that its path dependence is a separate function of direction of force, we can then expand the equation to describe such forces by simply running it again for each direction. Clunky, but it can work.
    I was considering gravity, air density, crosswinds, temperature, etc. all part of the "environment," which represents the function. Granted, that function will be tailored to the specific instance of when a specific ball was hit. But, I understand the value in both generalizing, and in treating these factors as discrete systems.

    I guess my mistake, here, is that I'm trying to back in to the quantification of a specific unit of work by building an equation to fit the measurement, rather than assembling equations that will accurately predict the outcome of a proposed unit of work. Also, I've been performing a self-imposed calculus refresher, this week, so the topic was ripe for me to apply my (mis)conceptions.

    Side note, I watched this lecture, the other day. Basic stuff, but way cool:

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    The only reason your method doesn't quite work is because some environmental factors are path dependent on earth and some are not.

    Technically, given an absolute vacuum and an unlimited amount of calculation time, your method of taking the function as an integral WOULD work, but it quickly becomes impractical to use for any degree of accuracy on earth as it requires you to re-run it not only for each force, which the original equation does as well, but also for each possible direction that force is coming from as well.

    You are trying to cheat your way around physics with math. I like it. Just because it isn't practical for most uses, doesn't mean it isn't REALLY COOL MATH.

    :)

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    Quote Originally Posted by AcerTempest View Post
    The only reason your method doesn't quite work is because some environmental factors are path dependent on earth and some are not.

    Technically, given an absolute vacuum and an unlimited amount of calculation time, your method of taking the function as an integral WOULD work, but it quickly becomes impractical to use for any degree of accuracy on earth as it requires you to re-run it not only for each force, which the original equation does as well, but also for each possible direction that force is coming from as well.

    You are trying to cheat your way around physics with math. I like it. Just because it isn't practical for most uses, doesn't mean it isn't REALLY COOL MATH.

    :)
    That's the trouble with fundamental physics: it sounds easy until it comes to application. Especially Earthbound stuff where the variables are numerous and prediction accuracy can be of a low percentage. But cool math is cool math and without it we'd still be shouting at each other in coffee shops.
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