The torque transmitted to an induction motor is a func-

PROBLEMS 203 s T 4 8 10 3 4 2 2 6 1 FIGURE P7.32 Torque transmitted to an inductor as a function of slip. 400 800 1,200 10,000 20,000 Total Minimum Lift Friction V D FIGURE P7.33 Plot of drag versus velocity for an airfoil. F F x FIGURE P7.34 Roller bearings. where n = revolutions per second of rotating stator speed and n R = rotor speed. Kirchhoff’s laws can be used to show that the torque expressed in dimensionless form and slip are related by T = 15s1 − s 1 − s4s 2 − 3s + 4 Figure P7.32 shows this function. Use a numerical method to determine the slip at which the maximum torque occurs. 7.33 The total drag on an airfoil can be estimated by D = 0.01σ V 2 + 0.95 σ W V 2 Friction Lift where D = drag, σ = ratio of air density between the flight altitude and sea level, W = weight, and V = velocity. As seen in Fig. P7.33, the two factors contributing to drag are affected differently as velocity increases. Whereas friction drag in- creases with velocity, the drag due to lift decreases. The com- bination of the two factors leads to a minimum drag. a If σ = 0.6 and W = 16,000, determine the minimum drag and the velocity at which it occurs. b In addition, develop a sensitivity analysis to determine how this optimum varies in response to a range of W = 12,000 to 20,000 with σ = 0.6. 7.34 Roller bearings are subject to fatigue failure caused by large contact loads F Fig. P7.34. The problem of finding the location of the maximum stress along the x axis can be shown to be equivalent to maximizing the function: f x = 0.4 √ 1 + x 2 − 1 + x 2 1 − 0.4 1 + x 2 + x Find the x that maximizes f x. 7.35 In a similar fashion to the case study described in Sec. 7.4, develop the potential energy function for the sys- tem depicted in Fig. P7.35. Develop contour and surface FIGURE P7.35 Two frictionless masses connected to a wall by a pair of linear elastic springs. 1 2 F k b k a x 1 x 2 plots in MATLAB. Minimize the potential energy function to determine the equilibrium displacements x 1 and x 2 given the forcing function F = 100 N and the parameters k a = 20 and k b = 15 Nm. 7.36 As an agricultural engineer, you must design a trape- zoidal open channel to carry irrigation water Fig. P7.36. Determine the optimal dimensions to minimize the wetted perimeter for a cross-sectional area of 50 m 2 . Are the relative dimensions universal? 7.37 Use the function fminsearch to determine the length of the shortest ladder that reaches from the ground over the fence to the building’s wall Fig. P7.37. Test it for the case where h = d = 4 m.

7.38 The length of the longest ladder that can negotiate

the corner depicted in Fig. P7.38 can be determined by computing the value of θ that minimizes the following function: Lθ = w 1 sin θ + w 2 sinπ − α − θ For the case where w 1 = w 2 = 2 m, use a numerical method described in this chapter including MATLAB’s built-in capabilities to develop a plot of L versus a range of α’s from 45 to 135 ◦ . FIGURE P7.37 A ladder leaning against a fence and just touching a wall. FIGURE P7.38 A ladder negotiating a corner formed by two hallways. h d a q w 2 w 1 L w d ␪ ␪ FIGURE P7.36 205 P ART T HREE Linear Systems

3.1 OVERVIEW

What Are Linear Algebraic Equations? In Part Two, we determined the value x that satisfied a single equation, f x = 0. Now, we deal with the case of determining the values x 1 , x 2 , . . . , x n that simultaneously satisfy a set of equations: f 1 x 1 , x 2 , . . . , x n = 0 f 2 x 1 , x 2 , . . . , x n = 0 . . . . . . f n x 1 , x 2 , . . . , x n = 0 Such systems are either linear or nonlinear. In Part Three, we deal with linear algebraic equations that are of the general form a 11 x 1 + a 12 x 2 + · · · + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + · · · + a 2n x n = b 2 .. . .. . a n 1 x 1 + a n 2 x 2 + · · · + a nn x n = b n PT3.1 where the a’s are constant coefficients, the b’s are constants, the x’s are unknowns, and n is the num- ber of equations. All other algebraic equations are nonlinear. Linear Algebraic Equations in Engineering and Science Many of the fundamental equations of engineering and science are based on conservation laws. Some familiar quantities that conform to such laws are mass, energy, and momentum. In mathematical terms, these principles lead to balance or continuity equations that relate system behavior as represented