Advanced Math

1 -3 2 -6 If T is defined by T(x)=Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A-| 0 1-2 | and b-1-26 5-16 10 Find a single vector x whose image under T is b

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EXPERT ANSWER 6 . Let the pre-image of (-6,-26,-2)T be x = (a,b,c)T where a,b, c are arbitrary real numbers. Then T(x) =Ax = (-6,-26,-2)T or, (a-3b+2c, b-2c, 5a-16b+10c)T = (-6,-26,-2)T . Hence, a-3b+2c = -6 …(1), b-2c = -26…(2) and 5a-16b+10c= 10…(3). The augmented matrix of the above linear system of equations is M (say)= 1 -3 2 -6 …

1 -3 2 -6 If T is defined by T(x)=Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A-| 0 1-2 | and b-1-26 5-16 10 Find a single vector x whose image under T is b Read More »

The following data apply to a 125-kW, 2300-V, three-phase, four pole, 60-Hz squirrel cage induction motor: Stator-resistance between phase terminals = 2.23 Ohm No-load test at rated frequency and voltage: Line current = 7.7 A Three-phase power = 2870 W Blocked-rotor test at 15 Hz: Line voltage = 268 V Line current = 50.3 A Three-phase power = 18.2 kW a. Calculate the sum of rotational and core losses. b. Calculate the equivalent-circuit parameters in ohms- Assume that X 1 = X 2. You can use the cantilever approximation. c. Compute the stator current, input power and power factor, output power and efficiency when this motor is operating at rated voltage and frequency at a slip of 2.95 percent.

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R-1 m-1 -0.12 -0.011 2 7. In each of the following 10 questions, pick up the best from the underlined five options. (1) A dynamical system with double poles at tj exhibits in (A) instability (B) marginal stability; (C) asymptotic stability (D) sustained oscillation (E) not the above. (2) Let D denote the derivative operator D=d/dt. Suppose a linear time-invariant (LTI) system G described as (D” + a,D*- + a,D* *+..+a, D+ a, y = (b,D” + b,D”- +…+bD+ b, u has zero initial conditions, and outputs y(t)= e. sint +2e cos 2t when subject to a unit-pulse u impulse. What is the value of n? (A) 2 (B) 3 (C) 4 (D) 6 (E) the information is not enough to determine the answer. Y(S) (3) As for a second-order system G(s) = its rising time and settling R(S) s? +2E0,5 +02 time, as we know, are mostly concerned with the natural frequency 0,. Which of the following does not properly make a sense to this phenomenon? (A) Higher-frequency vibration causes a higher damping (B) Lower natural frequency means lower stiffness, which in turns suggests the system be “soft”, and a “soft” system to an excitation takes longer time to settle down (C) Higher natural-frequency means higher stiffness, implying that the system is “hard”, and a hard system responses sharply to an impact (D) Higher-frequency vibration takes shorter time to cross a preset value (E) quick vibration quickly stops. (4) A simple harmonic motion governed by j+oʻy=0 with specified initial conditions being y(0)= y, and y(0)=v, has the general solution: y(t)=c, cos mt +c, sin ot, where two constants and c, are dependent on y, and v, . The value of a determines the frequency of the displacement y. As o is getting smaller, the displacement y is getting closer to DC level, and at the limit it becomes a constant y(t)=c,. On the other hand, as w=0, the governing equation becomes j=0, the solution of which is y(t) = y, +vet (an unstable system). Therefore, a seemingly contradiction happens. Such a contradiction happens because the above analysis did not (A) define the origin of time axis (B) distinguish existence from stability (C) specify whether the underlying application-range is an open set or a closed set (D) well tell DC signals from AC signals (E) know the limitation of mathematical analysis on physical systems.

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7. Show that asymptotical stability of linear time-invariant (LTI) systems is equivalent to quadratic Lyapunov stability. The proof needs to comprise the following two parts: (a) If a real matrix A has all eigenvalues in the left-hand plane, then the Lyapunov equation: A” P+PA+Q=0, Q>0, has a unique, positive-definite solution of P. (b) For any P=P”>0, Q=Q” >0, the Lyapunov equation: A’ P+PA+Q=0 implies that eigenvalues of the real matrix A are all in the left-hand plane.

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1. A particle moves through a region with a constant potential energy V 0 ​ . Show that the wavefunction Ψ(x)=Aexp(−αx), with α a real number, is a solution of the time-independent Schrödinger equation. What is the total energy of such a particle? 2. Consider an infinite potential well where V(x) ​ =0:x∈[0,D] =∞: elsewhere. ​ Show that the wavefunction Ψ(x)=sin( D nπx ​ ) describes a particle inside this potential well. You have to show that the wavefunction is both a solution of the time-independent Schrödinger equation, and also satisfies the necessary boundary conditions. [10] 3. A particle is described by the following wavefunction: Ψ 1 ​ (x)=−b(x 2 −a 2 ) 2 :0≤x≤a Ψ 2 ​ (x)=(x 2 −d 2 )−c:a≤x≤w Ψ 3 ​ (x)=0:x>w ​ (a) Using the continuity condition of an acceptable wavefunction at x=a , find c and d in terms of a and b . (b) Find w in terms of a and b . 4. A particle with zero (total) energy is described by the wavefunction Ψ(x) ​ =Acos( L 2πx ​ ):−L/4≤x≤L/4 =0: elsewhere. ​ (a) Determine the normalization constant A . (b) Calculate the potential energy of the particle. (c) What is the probability that the particle will be found between x=0 and x=L/8 ?

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Vehicles begin to arrive at a park entrance at 7:45 A.M. at a constant rate of six per minute and at a constant rate of four vehicles per minute from 8 A.M. on. The park opens at 8:00 A.M. and the manager wants to set the departure rate so that the average delay per vehicle is no greater than 9 minutes ( measured from the time of the first arrival until the total queue clears). Assuming D/D/1 queuing, what is the minimum departure rate needed to acheive this?

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Vehicles begin to arrive at a park entrance at 7:45 A.M. at a constant rate of six per minute and at a constant rate of four vehicles per minute from 8 A.M. on. The park opens at 8:00 A.M. and the manager wants to set the departure rate so that the average delay per vehicle …

Vehicles begin to arrive at a park entrance at 7:45 A.M. at a constant rate of six per minute and at a constant rate of four vehicles per minute from 8 A.M. on. The park opens at 8:00 A.M. and the manager wants to set the departure rate so that the average delay per vehicle is no greater than 9 minutes ( measured from the time of the first arrival until the total queue clears). Assuming D/D/1 queuing, what is the minimum departure rate needed to acheive this? Read More »

3-11. The 7/8 X 14 plate shown in Fig. 23-11. The holes are for 7/8-in (Ans. 10.54 in) bolts. bet in- 2in L PL x 14 FIGURE P3-11 3-12. The 6 X 4 X 1/2 angle shown has one line of 3/4-in bolts in each leg. The bolts are 4-in on center in each line and are staggered 2 in with respect to each other. 2 ir 34 in FIGURE P3-12 3-13. The tension member shown in Fig. P3-13 contains holes for 3/4-in Ø bolts. At what spacing, s, will the net area for the section through one hole be the same as a rup ture line passing though two holes? (Ans 3.24 in) 2 in FIGURE P3-13 3-14. The tension member shown in Figure P3-14 contains holes for 7/8-in bolts. At what spacing, s, will the net area for the section through two holes be the same as a rupture line passing through all three holes?

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