NAME: Massachuselts Institute of Technology 16.07 Dynamics Problem Set 10 Out date:Nov 17,2004 Due date:Nov 24.2004 Time Spent minutes Problem 1 Problem 2 Problem 3 (Optional) Study Time Trn in each problem separate sheetsso
NAME : . . . . . . . . . . . . . . . . . . . . . Massachusetts Institute of Technology 16.07 Dynamics Problem Set 10 Out date: Nov 17, 2004 Due date: Nov 24, 2004 Time Spent [minutes] Problem 1 Problem 2 Problem 3 (Optional) Study Time Turn in each problem on separate sheets so that grading can be done in parallel
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Problem 1 Part A The 10 kg plate is subjected to a force F = 20 N which is always perpendicular to the face of the plate. If the plate is originally at rest and in the x − z plane, determine its angular velocity after it has rotated one revolution. The plate is supported by ball-and-socket joints at A and B. N m m Hint: If we know the tensor of inertia about the orthogonal axis xyz, with origin at the center of mass, [I]xyz, we can calculate the tensor of inertia about about the orthogonal set x ′ y ′ z ′ , also with origin at the center of mass, [I]x′y ′z ′, using the following matrix transformation expression, [I]x′y ′z ′ = Q T [I]xyz Q where [I]x′y ′z ′ = Ix′x′ −Ix′y ′ −Ix′z ′ −Iy ′x′ Iy ′y ′ −Iy ′z ′ −Iz ′x′ −Iz ′y ′ Iz ′z ′ , [I]xyz = Ixx −Ixy −Ixz −Iyx Iyy −Iyz −Izx −Izy Izz , and Q = i ′ · i i′ · j i′ · k j ′ · i j′ · j j′ · k k ′ · i k′ · j k′ · k . Here, i, j, and k are the unit vectors in the direction of the xyz axes and i ′ , j ′ , and k ′ are the unit vectors in the direction of the x ′ y ′ z ′ . Part B For the same problem as in part A, determine the reactions at points A and B after the plate has turned one revolution
Problem2 A ded from G the body n ge 0.How e to the pr
Problem 2 Part A (Kleppner/Kolenkow) A gyroscope wheel is at one end of an axle of length l. The other end of the axle is suspended from a string of length L. The wheel is set into motion so that it executes uniform precession in the horizontal plane. The wheel has mass M and moment of inertia about its center of mass I0. Its spin angular velocity is ωs. Neglect the mass of the shaft and of the string. Find the angle β that the string makes with the vertical. Assume that β is so small that approximations like sin β ≈ β are justified. Part B We consider the body axes xyz, for an aircraft, to be those shown in the diagram: x is directed along the fuselage, y is to the right, in the wing’s plane, and z is perpendicular to x and y. We can generally assume that the aircraft is symmetric with respect to y = 0. However, in general, x and z will not be principal axes of inertia. In fact, we will typically get Ixz > 0 due to the presence of the tail
Write the general form of the ter of inertia in the axes ·m如oe去品e品aAa How would this torque be applie how that this torque depends onr- Problem 3 THIS PROBLE Problem 7/12fro Meriam and Kraige,Dyuamics,Fifth Edition
• Write the general form of the tensor of inertia in these axes. • Suppose we want to maintain a pure steady roll Ωx 6= 0, Ωy = Ωz = 0. What torque, if any, needs to be applied?. How would this torque be applied? • Suppose now that Ixz is really small and therefore can be neglected. What torque would need to be applied to do a pull up roll Ωx, Ωy 6= 0, while keeping Ωz = 0? How would this torque be applied? Show that this torque depends on Ix − Iy. Problem 3 THIS PROBLEM IS OPTIONAL AND STUDENTS THAT COMPLETE IT SUCCESSFULLY WILL OBTAIN ADDITIONAL CREDIT EQUIVALENT TO ONE HOMEWORK PROBLEM. Problem 7/126 from Meriam and Kraige, Dynamics, Fifth Edition Hint: Note that ω 2 0 is small compared to pω0. Also, β is small and therefore, sin β ≈ β