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systems analysis and design using matlab

Questions and Answers of Systems Analysis And Design Using MATLAB

31. Damped free vibrations can be modeled by a block of mass m that is attached to a spring and a dashpot as shown. From Newton’s second law of motion, the displacement x of the mass as a function
30. The current, i, in a series RLC circuit when the switch is closed at can be determined from the solution of the 2nd-order ordinary differential equation(ODE):where R, L, and C are the resistance
29. Determine the solution of the following differential equation that satisfies the given initial conditions. Plot the solution for ., , vS LR B A vR R = 0.4 i L = 0.08 t = 0 vS vS = 6 tBA iR L di
28. Determine the general solution of the differential equation:Show that the solution is correct. (Derive the first derivative of the solution, and then substitute back into the equation.)
27. A resistor R ( ) and an inductor L ( H) are connected as shown. Initially, the switch is connected to point A and there is no current in the circuit. At the switch is moved from A to B, so that
26. The velocity of a skydiver whose parachute is still closed can be modeled by assuming that the air resistance is proportional to the velocity.From Newton’s second law of motion the relationship
25. The Maxwell-Boltzmann probability density function is given by:where m (kg) is the mass of each molecule, v (m/s) is the speed, T (K) is the temperature, and J/K is Boltzmann’s constant. The
24. The spread of an infection from a single individual to a population of N uninfected persons can be described by the equation:with initial condition where x is the number of uninfected individuals
23. The rms value of an AC voltage is defined by:where T is the period of the waveform.(a) A voltage is given by . Show that and is independent of . (The relationship between the period T and the
22. Consider the parabolic sector shown in the previous problem. Show that the moment of inertia about the x axis, , is given by . The moment of inertia can be calculated by:
21. Show that the location of the centroid of the parabolic sector shown is given by. The coordinate can be calculated by:
20. A ceramic tile has the design shown in the figure. The shaded area is painted red and the rest of the tile is white. The border line between the red and the white areas follows the
19. The one-dimensional diffusion equation is given by:Show that the following are solutions to the diffusion equation.(a) , where A and B are constants.(b) , where A, B, C, and are constants.
18. Define x as a symbolic variable and create the symbolic expression:Plot S in the domain and calculate the integral .
17. Evaluate the following indefinite integrals
16. The parametric equations of an ellipsoid are:, , where and .Show that the differential volume element of the ellipsoid shown is given by:Use MATLAB to evaluate the integral of dV from to 0
15. A tracking radar antenna is locked on an airplane flying at a constant altitude of 5 km, and a constant speed of 540 km/h. The airplane travels along a path that passes exactly above the radar
14. The equation of a circle with its center is at and is given by, where R is the radius of the circle. Write a program in a script file that first derives the equation(symbolically) of the tangent
13. The mechanical power output P in a contracting muscle is given by:where T is the muscle tension, v is the shortening velocity (max of ), is the isometric tension (i.e., tension at zero velocity),
12. A box of mass m is being pulled by a rope as shown. The force F in the rope as a function of x can be calculated from the equations:where N and are the normal force and friction coefficient
11. A 120 in.-long beam AB is attached to the wall with a pin at point A and to a 66 in.-long cable CD. A load lb is attached to the beam at point B. The tension in the cable T and the x and y
10. Consider the two ellipses in the x y plane given by the equations:and(a) Use the ezplot command to plot the two ellipses in the same figure.(b) Determine the coordinates of the points where the
9. The relation between the tension T and the steady shortening velocity v in a muscle is given by the Hill equation:where a and b are positive constants and is the isometric tension, i.e., the
8. A water tank has the geometry shown in the figure (the upper section is a cylinder with radius r and height h, and the lower section is a cone with radius r and a height of 2r). Determine the
7. In rectangular coordinates the equation of the hyperbola shown in the figure is given by:(a) Use MATLAB to show that in parametric form the equation of the hyperbola can be written as:and(b) Make
6. Use the commands from Section 11.2 to show that:(a)(b)
5. Use the commands from Section 11.2 to show that:(a)(b)
4. Define x as a symbolic variable.(a) Derive the equation of the polynomial that has the roots , ,, , and .(b) Determine the roots of the polynomial:by using the factor command.
3. Define x and y as symbolic variables and create the two symbolic expressions and Use symbolic operations to determine the simplest form of . Use the subs command to evaluate the numerical value of
2. Define y as a symbolic variable and create the two symbolic expressions and Use symbolic operations to determine the simplest form of each of the following expressions:(a) (b) (c)(d) Use the subs
1. Define x as a symbolic variable and create the two symbolic expressions and Use symbolic operations to determine the simplest form of each of the following expressions:(a) (b) (c)(d) Use the subs
22. A ball thrown up falls back to the floor and bounces many times. For a ball thrown up in the direction shown in the figure, the position of the ball as a function of time is given by:The
21. The stress fields near a crack tip of a linear elastic isotropic material for mode II loading are given by:For ksi plot the stresses (each in a separate figure) in the domain and in. Plot the
20. The geometry of a ship hull (Wigley hull) can be modeled by the equation:where x, y, and z are the length, width, and height, respectively. Use MATLAB to make a 3-D figure of the hull as shown.
19. The Verhulst model, given in the following equation, describes the growth of a population that is limited by various factors such as overcrowding and lack of resources:where is the number of
18. The deflection w of a clamped circular membrane of radius subjected to pressure P is given by (small deformation theory):where r is the radial coordinate, and , where E, t, and are the elastic
17. The equation for the streamlines for uniform flow over a cylinder is where is the stream function. For example, if , then . Since the equation is satisfied for all x, the x axis is the zero ( )
16. A high-pass filter passes signals with frequencies that are higher than a certain cutoff frequency.In this filter the ratio of the magnitudes of the voltages is given by:where is the frequency of
15. In the solution of elasticity problem of a normal point load applied to the surface of a half plane that was solved by Boussinesq in 1878, the stresses and are given by:and where is Poisson’s
14. An RLC circuit with an alternating voltage source is shown. The source voltage is given by, where , in which is the driving frequency.The amplitude of the current, I, in this circuit is given
13. Consider steady-state vibration of a friction-free spring-mass-damper system subjected to harmonic applied force.The normalized amplitude of the mass is given by:where is the frequency ratio, and
12. Plank’s distribution law gives the blackbody emissive power (amount of radiation energy emitted) as a function of temperature and wavelength:where W m4/m2, mK, T is the temperature in degrees
11. Molecules of a gas in a container are moving around at different speeds.Maxwell’s speed distribution law gives the probability distribution as a function of temperature and speed:where M is the
10. A monochromatic light that passes through a slit produces on a screen a diffraction pattern consisting of bright and dark fringes. The intensity of the bright fringes, I, as a function of can be
9. Make a plot of the ice cream cone shown in the figure.The cone is 8 in. tall with a 4-in. diameter base. The ice cream at the top is a 4-in. diameter hemisphere.A parametric equation for the cone
8. Make a 3-D surface plot of the function in the domain and .
7. Make a 3-D mesh plot of the function , where in the domain and .
6. Make a 3-D surface plot of the function in the domain and .
5. Make a 3-D surface plot of the function in the domain and .
4. Make a 3-D surface plot of the function in the domain and .
3. The ladder of a fire truck can be elevated(increase of angle ), rotated about the z axis (increase of angle ), and extended (increase of r). Initially the ladder rests on the truck ( ,, and m).
2. A staircase of height h is modeled by the parametric equations:where , , and m is the staircase height.Make a 3-D plot (shown) of the staircase.(Create a vector t for the domain 0 to , and use the
1. The position of a moving particle as a function of time is given by:Plot the position of the particle for s.
42. The velocity, v, of an object that falls freely due to the Earth gravity can be modeled with the equation:where m is the mass of the object, m/s2, and k is a constant. Solve the equation for v
41. Growth of many organisms can be modeled with the equation:where is the mass of the organism, is the assumed maximum mas, and k is a constant. Solve the equation for days, given kg1/4/day, kg and
40. An RL circuit includes a voltage source , a resistor , and an inductor H, as shown in the figure. The differential equation that describes the response of the circuit is where is the current in
39. The population growth of species with limited capacity can be modeled by the equation:where N is the population size, is the limiting number for the population, and k, r, and are constants. The
38. An airplane uses a parachute and other means of braking as it slows down on the runway after landing. Its acceleration is given by m/s2. Since, the rate of change of the velocity is given
37. The sudden outbreak of an insect population can be modeled by the equation The first term relates to the well-known logistic population growth model where N is the number of insects, R is an
36. A water tank shaped as a cone ( m, m) has a circular hole at the bottom( mm), as shown. According to Torricelli’s law, the speed v of the water that is discharging from the hole is given
35. The growth of a fish is often modeled by the von Bertalanffy growth model:where w is the weight and a and b are constants. Solve the equation for w for the case lb1/3, day–1, and lb. Make sure
34. Use a MATLAB built-in function to numerically solve:for with Plot the solution.
33. Use a MATLAB built-in function to numerically solve:for with In one figure plot the numerical solution as a solid line and the exact solution as discrete points (10 equally spaced points).Exact
32. Use a MATLAB built-in function to numerically solve:for with In one figure plot the numerical solution as a solid line and the exact solution as discrete points.Exact solution: .
31. Use a MATLAB built-in function to numerically solve:for with Plot the numerical solution.
30. The Fresnel integrals are:and Calculate and for (use spacing of 0.05). In one figure plot two graphs—one of versus x and the other of versus x. In a second figure plot versus .
29. The orbit of Mercury is elliptical in shape, with km and km. The perimeter of an ellipse can be calculated by where . Determine the distance Mercury travels in one orbit. Calculate the average
28. A cross-sectional area has the geometry of half an ellipse, as shown in the figure to the right. The coordinate of the centroid of the area can be calculated by:where A is the area given by , and
27. To estimate the surface area and volume of a football, the diameter of the ball is measured at different points along the ball. The surface area, S, and volume, V, can be determined by:and Use
26. An approximate map of Lake Erie is shown in the figure. Use numerical integration to estimate the area of the lake. Make a list of the width of the lake as a function of x. Start with mi and use
25. The variation of gravitational acceleration g with altitude y is given by:where km is the radius of the Earth, and m/s2 is the gravitational acceleration at sea level. The change in the
24. The length of a curve given by a parametric equation , is given by:The cardioid curve shown in the figure is given by:and with . Plot the cardioid with and determine the length of the curve.
23. The flow rate Q (volume of fluid per second)in a round pipe can be calculated by:For turbulent flow the velocity profile can be estimated by: . Determine Q for in.,, in./s.
22. The electric wire that connects the house to the pole has the shape of a catenary:By using the equation:determine the length of the wire.
21. Use numerical integration to approximate the size of the shaded area shown in the figure. Create a vector with values of x from 1 through 10 and estimate the corresponding y coordinate. Then,
20. A rubber band is stretched by fixing one end pulling the other end. Measurements of the applied force at different displacements are given in the following table:Determine the work done by the
19. The speed of a race car during the first 7 s of a race is given by:Determine the distance the car traveled during the first 7 s.
18. Use MATLAB to calculate the following integrals:(a) (b)
17. Use MATLAB to calculate the following integrals:(a) (b)
16. A 108-in.-long beam AB is attached to the wall with a pin at point A and to a 68-in.-long cable CD. A load lb is attached to the beam at point B. The tension in the cable T is given by:where L
15. An RLC circuit with an alternating voltage source is shown. The source voltage is given by , where , in which is the driving frequency. The amplitude of the current, I, in this circuit is given
14. A prismatic box with equilateral triangular base is made from a equilateral triangular sheet with sides s by cutting off the corners and folding the edges along the dashed lines. For in., use
13. Using MATLAB’s built-in function fminbnd, determine the dimensions (radius r and height h)and the volume of the cylinder with the largest volume that can be made inside of a cone with a radius
12. A flat rectangular sheet of metal that is 70 in. wide and 120 in. long is formed to make a container with the geometry shown in the figure. (Additional flat metal pieces are attached at the
11. Using MATLAB’s built-in function fminbnd, determine the minimum and the maximum of the function
10. For fluid flow in a pipe, the Colebrook–White (or Colebrook) equation gives a relationship between the friction coefficient,f, and the Reynolds number:where k/d is the pipe relative roughness.
9. A series RLC circuit with an AC voltage source is shown. The amplitude of the current, I, in this circuit is given by:where in which is the driving frequency; R and C are the resistance of the
8. An estimate of the minimum velocity required for a round flat stone to skip when it hits the water is given by (Lyderic Bocquet, “The Physics of Stone Skipping,” Am. J. Phys., vol. 71, no. 2,
7. The van der Waals equation gives a relationship between the pressure p(atm), volume V (L), and temperature T (K) for a real gas where n is the number of moles, (L atm)/(mol K) is the gas constant,
6. The position s of the slider as a function of in the crank-slider mechanism shown is given by:Given in., in., and in., determine the angle , when in. (There are two solutions.)
5. The area A of a circle segment is given by:Determine the angle (in degrees) if in. and in2.
4. Determine the positive roots of the equation .
3. Determine the three roots of the equation .
2. Determine the solution of the equation .
1. Determine the two solutions of the equation between and .
35. The transmission of light through a transparent solid can be described by the equation:where is the transmitted intensity, is the intensity of the incident beam, is the absorption coefficient, L
34. When rubber is stretched, its elongation is initially proportional to the applied force, but as it reaches about twice its original length, the force required to stretch the rubber increases
33. Curve-fit the data from the previous problem with a third-order polynomial.Use the polynomial to estimate y at . Make a plot of the points and the polynomial.
32. The relationship between two variables y and x is known to be:The following data points are given:Determine the constants a and b by curve-fitting the equation to the data points. Make a plot of
31. Use the data from Problem 30 for the following:(a) Fit the data with linear interpolation. Estimate the concentration at. Make a plot that shows the data points and curve made of interpolated

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