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A problem statement, regarding dynamics of a BODY, contains specification of the acceleration of the BODY at the instant of initiation of its event as a "given." Thus the author implies, by some physical means or manner, "at the commencing instant of an event" the acceleration of the BODY is determined. Below is a discussion of how, physically, that "initial condition" might be known.
Conditions and the "Pre-Event:" Regarding acceleration in general. If at an instant in time, a BODY has an acceleration (non-zero) then in the next instant the BODY moves. The BODY is not where it was an instant later. Further, from its initial location, should a BODY not change location for its event, then it had no acceleration as its initial conditon.
The "physical/mathematical" meaning of event is important. Our discussion addresses the "event" of a BODY. The event initiates at an instance in time with the BODY characteristics at that time - its "initial conditions." The event procedes in time (for its duration, epoch or era) then, abruptly the event terminates. Interestingly, to state the "initial condition" of a BODY, requires consideration of another event, specifically, the "end" of the BODY event immediately prior the event under investigation.
Cases: Below are descriptions of physical circumstances "prior to our event" that assure a known initial acceleration of our BODY for its event.
♦ BODY above Earth If a BODY is suspended above Earth by the tension in a "constraining cord," and the event is described to commence upon, sever of the cord - the initial acceleration is rightly stated as the surface of Earth value, go,E .
♦ Vehicle Accelerated In this case a mass hangs on a cord attached to the rear-view mirror of an auto. While stationary, the cord alignment is vertical. Moments later the auto is in motion. Sketch (a) shows a condition of the pendant during motion.
Calculate the acceleration of the car at the instant sketch (a) applies.It is logical to model the pendant as a BODY and to apply Newton's 2nd Law. A first step to apply Newton's Method is to "identify a system" then to "isolate" it. This is a mental, pencil and paper activity. To proceed in application
The "free-body-diagram" provides a vector basis for writing Newton's 2nd Law for the mass. The forces are the tension of the cord and the force of gravity.
|(1) The length of a radial arc|
We write the cord tension in magnitude/direction form. The angle of the cord with the vertical is notated (for now) as a function of time.
|(2) to "find" the number.|
For our physical event, the angle of inclination, β, is seen to be constant in time.
|(3) to "find" the number.|
Next we write the acceleration in component form, remove "time-dependence" from the angle, β then scalar multiply the(3) by the unit vectors I, then K. Two scalar equations result to be solved:
|(4) to "find" the number.|
The "third" equation of Eqn-4, has Az = 0 because by observation the elevation of the BODY does not change. The BODY does not move in the "z" direction that component of its acceleration has magnitude, zero.
|(5) to "find" the number.|