Exam 05: Chapters 10 and 11

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1 Name: Exam 05: Chapters 10 and 11 Select and solve five of the following problems to the best of your ability. You must choose two problem from each column, and a final problem at your own discretion. Indicate below which five problems you wish to have graded. If you do not explicitly mark a problem to be scored, it will not be scored. If you have worked on more than five problems, select only five to be graded. I will not choose for you. Choose At Least Two Grade this one? Score Choose At Least Two Grade this one? Score Problem 01 /15 Problem 06 /30 Problem 02 /25 Problem 07 /25 Problem 03 /15 Problem 08 /25 Problem 04 /15 Problem 09 /25 Problem 05 You may use your calculator and the attached formula sheet. Read and follow the directions carefully. Solve using the method required by the problem statement. If you are not explicitly required to use a specific technique, please be sure to show sufficient work so that your method is obvious. Show all your work. Work as neatly as you can. If you need scratch paper, blank sheets will be provided for you. It is permissible to use your calculator to solve a system of equations directly. If you do, state this explicitly. Express your answer as directed by the problem statement, using three significant digits. Include the appropriate units. EXAM 05! PAGE 1

2 Problem 01 Use direct integration to determine the moment of inertia with respect to the x-axis: (Hint: A horizontal strip is the easiest area increment.) dy x EXAM 05! PAGE 2

3 Problem 02 A. Locate the centroid y of the channel s crosssectional area. B. Determine the moment of inertia with respect to the x axis passing through the centroid. y A3 x C x A1 A2 EXAM 05! PAGE 3

4 Problem 03 Use direct integration to determine the mass moment of inertia Ix with respect to the x-axis. The density of the zinc alloy used is 6800 kg/m 3. For a disk-shaped mass increment dm: HINT: If y 2 = 50x, then to be dimensionally consistent, the constant must have units!! (y mm) 2 = (50mm)(x mm). Be doubleplus extra careful with your units. EXAM 05! PAGE 4

5 Problem 04 The pendulum consists of a 3-kg slender rod attached to a 5-kg thin plate. A. Determine the location ȳ of the center of mass G of the pendulum. B. Find the mass moment of inertia of the pendulum about an axis perpendicular to the page and passing through G. ȳ y1 G1 d1 d2 y2 G2 EXAM 05! PAGE 5

6 Problem 05 Determine the moment of inertia Iz of the frustum of the cone which has a conical depression. The alpha bronze used to make the part has a density of 8470 kg/m 3. Cone 1: Complete solid Cone 2: Negative tip 0.8 h 0.2 Cone 3: Negative depression 0.4 EXAM 05! PAGE 6

7 Problem 06 Determine the angles θ for equilibrium of the 4-lb disk using the principle of virtual work. Neglect the weight of the rod. The spring is unstretched when θ = 0 and always remains in the vertical position due to the roller guide. Disk at A: Spring at C: ya yc EXAM 05! PAGE 7

8 Problem 07 The spring has a torsional stiffness of k = 300 N m/rad and is unstretched when θ = 90. Use the method of virtual work to determine the angles θ when the frame is in equilibrium. Ignore the masses of the frame members. HINT: MB for the torsion spring is k( α), where α is the angle at B. You ll need the relationship between angles θ and α. The trig equation factors easily, for two solutions. Torque at A: α Torsion spring at B: Virtual Work: EXAM 05! PAGE 8

9 Problem 08 The spring is unstretched when θ = 45 and has a stiffness of k = 1000 lb/ft. Use the potential energy method to determine the angle θ for equilibrium if each of the cylinders weighs 50 lb. Neglect the weight of the members. Spring at E: Masses at B and C: Potential Function: xe yb = yc EXAM 05! PAGE 9

10 Problem 09 The uniform bar AB weighs 100 lb. If both springs DE and BC are unstretched when θ = 90. Both springs always remain in the horizontal position due to the roller guides at C and E. A. Determine the angle θ for equilibrium using the principle of potential energy. B. Evaluate the stability of the equilibrium position. HINT: Units!! All inches or all feet pick one! Trig equation solves easily by factoring, yields two values for θ. Potential Energy of Bar: xd xb yg Potential of Springs B and D: Potential Function: Stability of Equilibrium: Neither position is stable! EXAM 05! PAGE 10

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