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Dynamics: Practice Final – 3-D Kinematics of Rigid Bodies Show all work for partial credit. Use additional paper if necessary. 1. Find the components of the inertia matrix for the compound bar shown. The bar has a mass of 3 kg/m. (A slender bar has a moments of inertia about its center 1/12 m·l2, its moment of inertia about its long axis is zero, and its products of inertia about its center are zero). Ixx = ___________________ kg·m2 Izz = ___________________ kg·m2 Iyz = ___________________ kg·m2 2. A certain rigid body has an inertia matrix of [I], given below. What is the body’s moment of inertia if it is spun about an axis in the direction û , also given below? ⎡ 20 0 − 5⎤ ⎡3 / 5 ⎤ ⎥ ⎢ 2 [I]= ⎢ 0 15 − 5⎥ kg·m û = ⎢⎢ 0 ⎥⎥ ⎢⎣− 5 − 5 10 ⎥⎦ ⎢⎣4 / 5⎥⎦ 3. A certain rigid body has an inertia matrix of [I], given below. What is the body’s angular momentum if r it is spun with angular velocity ω , also given below? ⎡ 20 0 − 5⎤ ⎡1⎤ r ⎢⎥ ⎥ ⎢ 2 ω = ⎢1⎥ rad/sec [I]= ⎢ 0 15 − 5⎥ kg·m ⎢⎣− 5 − 5 10 ⎥⎦ ⎢⎣1⎥⎦ 4. A certain rigid body has an inertia matrix of [I], given below. What is the body’s rotational kinetic r energy if it is spun with angular velocity ω , also given below? ⎡ 20 0 − 5⎤ ⎡1⎤ r ⎢⎥ ⎢ ⎥ 2 ω = ⎢1⎥ rad/sec [I]= ⎢ 0 15 − 5⎥ kg·m ⎢⎣− 5 − 5 10 ⎥⎦ ⎢⎣1⎥⎦ 5. A uniform 40 kg, 2 meter rod floating in space and aligned along the z-axis experiences two forces applied at its ends. Force A is 30 N in the x-direction, and force B is 20 N in the y-direction. Find the following values at this instant: • The inertial linear acceleration of the center of mass • The angular acceleration about the center of mass MAE-2102 1 Ver. D, Rev. 1 D Dynamic cs: Practicce Final – 3-D Kinnematics of o Rigid Bodies B 6. Att the instant sho own, the towerr crane is rotatiing about the z axis with an angular velocity v ω1 = 0.25 0 rad/s, whicch is increasingg at 2 0.66 rad/s . The boom b OA is rottating downwarrd with an anguular vellocity ω2 = 0.4 4 rad/s, which is increasing att 0.8 rad/s2. • • • ω&1 =0.6 rad/s2 B Determine th he angular veloocity of the booom Calculate thee velocity of pooint B located at the top of thhe boom. Find the acceleration of pooint B. ω& 2 =0.8 rad/s2 7. own, the base of o the robotic arm a is turning Att the instant sho aboout the z axis with w an angularr velocity of ω1 =4 rad/s, whhich is increasing at ω&1 = 3 raad/s2. Also, thee boom seggment BC is ro otating at ω BC = 8 rad/s, whicch is increasingg vC=1 m/s at ω& BC = 2 rad/s2. The gripper is retracting att a constant ratte vC =1 m/s. • • C Caalculate the velocity of point C. (exxtra) Find the acceleration a off point C. 8. A 4 kg thin ring of o diameter 1 meter m hangs froom a nail in a wall. w If the rinng is started roccking in the plaane of the ring about the nail with a small angular a displaceement, what is the perriod of oscillattion of the ringg? E-2102 MAE 2 Ver. D, Rev. 1