Motion 5.3 2
Apr 17, 2017 Motion 5.3.2 addresses some stability issues, including when scaling shapes from or to zero, working with duplicated rigged filters and behaviors, moving the cursor across markers during playback, and using keyboard shortcuts while a keyframe is selected. Apr 15, 2017 Apple Motion 5.3.2 – Create and customize Final Cut Pro titles, transitions, and effects. April 15, 2017 Apple Motion is designed for video editors, Motion 5 lets you customize Final Cut Pro titles, transitions, and effects. 2017 iMac 27' 8Gb Graphics 40Gb RAM 512Gb SSD i7 4,2Ghz Sierra 10.12.6 and Motion 5.3.2 2013 MacBook Pro Retina 13' with the same OS: Sierra 10.12.6 and Motion 5.3.2 Other things that I tried: - Creating a simple project with a white square in the middle and tried to export to EXRs.
5.3.2 A Differential Equation for Spring Motion
By Newton's Second Law of Motion, the total force on the moving object of mass (m) is also its mass times its acceleration. Thus, we have a relationship between the displacement (x) and its second derivative:
Some features of this equation should look familiar and some should appear new. First, under the influence of gravity, our bouncing mass is a falling body — but with a second force acting on it, the spring. When the falling body had no other force acting on it, our differential equation model expressed the second derivative of the position function as a constant. Here the second derivative is proportional to the position function.
Motion 5.3.2 Torrent
Second, as was the case with the falling body, we should expect to undo differentiation twice to find (x) as a function of (t). But, unlike the falling body model, it is not at all clear how to do that. We might also expect that two 'undifferentiation' steps will need two initial conditions, probably an initial velocity and an initial position.
Motion 5.3
Finally, note that the phrase 'proportional to the unknown function' has appeared before — in natural population growth. What's different here is that it is the second derivative, not the first, that is proportional to the unknown function. Of course, an exponential function, say, (f(t)=e^{rt}), has a second derivative that is proportional to the function itself, but it seems unlikely that such a function could tell us anything about bouncing up and down on a spring. Let's rule out exponential functions as solutions to the differential equation right away.
Activity 1
Calculate the second derivative of
Show that (f,') is proportional to (f) and that the proportionality constant must be positive, no matter what sign (r) has — as long as (r) is not (0).
Explain why (f) cannot be a solution of an equation of the form
Motion 5.3.2
We have just eliminated a promising candidate — promising mathematically, but not physically — for a solution to the differential equation
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In fact, as we will see, the functions we need to model spring motion and other oscillations and vibrations are sines and cosines. We turn now to investigating these fundamental functions.