Threshold

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threshold is the physical state of conversion from an thing to different physical state, a physical condition to be met, often mention that the mathematical equivalent of stagnation. Therefore, using the critical state of most and minimum physical solution, solving physics has transform the maximum essential value of a means. But I believe namely using the critical state to solve the most value should be wary, we have to premier differentiate between two states.

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relationship analysis condition 1 condition 2 condition 3 Conclusion algebraic analysis aboard the relationship between the critical worth of 1 is above the relationship between the two conditions are diagnosed and argued below.

threshold

?? 1. Two states exist at the critical point and get the most value?? Example 1 shown in Figure 1, mass m of the pellet with initial velocity v0 by the smooth begin from point A circular groove on the slippery pathway, find the ball via the highest point B in a circular film of the minimum initial velocity v0. Figure 1?? Analysis because the track groove is flat, the plenary process system mechanical energy protection. The ball in the A-point total for?? Ek = (1 / 2) mv02,?? At point B of the total energy?? E = ( 1 / 2) mv02 +2 mgR, then (1 / 2) mv02 = (1 / 2) mvB2 +2 mgR, ①?? evidently ① where v0 to make the minimum, requires vB minimum, so we quest the minimum value v0 into the minimum for the sake of vB. But only along to ① vB neatness we can no make sure the minimum. To determine the minimum value should too be considered vB questions provided in the conditions: the ball via point B in a circular film without falling and falling exists between the critical speed, which must be greater than or equal to the vB critical speed, ball will not fall, so speed is the critical point B without the ball falling through the minimum speed, thus solving the minimum initial velocity ball as the sake of a small ball into a circular orbit along a vertical circular motion of the critical speed. ?? Solution of the ball in a circular film to the maximum point B when the coerce is: their orbits at gravity mg and the character of oppression N, the two together for the ball to invest a circular motion the centripetal force, that is?? F Heart = mg + N, so (mvB2 / R) = mg + N, ② ① to ② eq we can obtain?? concluded the most value for such problems as the most value to obtain the critical point, we can use the threshold with the most the relationship between the two values, based on the object by a critical point in the process of alteration followed the decrees of physics to ascertain out the threshold, and thus find the best value. This is when using some additional method to solve complex or simply seeking not appear the most value problems, can not say it is not a shortcut. Condition 2?? 2. Two states exist, but the most critical point value is not achieved?? Example 2 shown in Figure 2, a mass of 10kg of objects by one class to the left of the constant force F = 10N role, starting from point A to the initial speed v0 = 10m / s to the right along a smooth horizontal slip, find objects within minutes in the 8s maximum displacement (to the right is assured). Figure 2?? Analysis is clear that the problem exists in the speed of a critical point,9112.org, ie v = 0 points. According to the formula vt2-v02 = 2as obtainable s Pro = 50m. It should be eminent that v = 0 is the period required to 10s, when the period is less than 10s, the displacement s is one increasing function, that is?? Smax = 10 × 8-( 1 / 2) × 1 × 82 = 48m, so s Pro ≠ smax. ?? Concluded the most value for such problems as the most critical point value is not achieved, as the threshold can not be the most value, merely according to the conditions by means of the threshold question in determining the most value condition 3?? 3. The most critical point there is no value, but an endless value of trend changes in the threshold?? Example 3 shown in Figure 3, in two parallel curtate rails, spacing is L, the electromotive force is E, all the resistance R, metal rods ab mass m, friction between it and the slide element μ. The entire circulate in the catching induction B of the uniform fascinating field. When switch S is closed, ab sliding bar to the right begin from repose. Asked if the rate of ab wand to accomplish maximum value? Like to accomplish, find its maximum value. 3?? Analysis is a quite classic technicians, electromagnetism and comprehensive title, the general solution are such that: As the wand ab start is to do the movement haste decreases, so that the haste becomes naught when the maximum speed, which speed approaches its maximum by the critical point, using the retinue formula to count the maximum speed.?? mgμ = BIL,? ? mgμ = BL (E-BLv / R),?? vmax = (E / BL) - (mgμR/B2L2). ?? Here, we stick ab along to changes in the whole process followed by physical laws, mathematical method to enumerate the speed of rod ab statement over period, and then discuss the most value, because it is variable force, so we must account the following differential equation, that is?? a = (dv / dt) = (BLI-f / m), ③?? I = (E '/ R) = (E-BLv / R),antigutkha, ④ the ③ ④ into available?? (dv / dt) + (B2L2/mR) v = (BL / mR)-gμ, ⑤ ⑤ type is a first-order straightforward differential equations, the general solution ⑥ When at t = 0, v = 0, we can get into ⑥?? C =- (BLE-gμmR) / (B2L2), so?? discussion is monotonically increasing function v is infinite towards its critical value: (E / BL) - (mgμR / B2L2), so there is no maximum. Only as t → ∞, v can take the limit (threshold), which is often said that the Terminal Velocity. Conclusion?? Conclusion whatever the subject of numerous samples, such as when the capacitor dictate and discharge voltage on the capacitor, rain resistance f =- kv by the speed, I have studied the campaign of them have one common feature: the changes in the demand The physical changes are connecting feedback and follow first-order straightforward differential equations. Because of the critical point and effortless to determine, often mistakenly regard it as the best value, best value does not actually exist, only in the circumstance of similar values ​​that most of its existence and equal to its critical value. ?? In synopsis, I believe that the critical point is decisive by the physical decrees of a state, it can be to meet the physical laws of the state to determine, is an objective reality. The best value solution, strictly speaking, should be beneath certain conditions and physical laws under a change control process, this process can achieve the most value is obtained at the critical point, a comprehensive analysis of it to meet the conditions and the following The laws of physics, a agreeable clutch of the changes in physical characteristics, but too follow definite mathematical principles. In some cases (as in Example 1), the critical value and the very best value we can for the most critical point value, to obtain a multiplier achieve. In other cases (such as patients 2 and 3), the most value and has no necessary linkage between the critical value. Therefore, the best value solution process, we can only find the critical points as a direction of seeking the best value, not capable to solve it with the same value up. Mathematical threshold in the test for independence, the value determined according to the actual results of the experiment. 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