The University of Melbourne 436-291 Engineering Mechanics Tutorial Eleven Instantaneous Centre and General Motion Part A (Introductory) 1. (Problem 5/93 from Meriam and Kraige - Dynamics) For the instant represented in Figure 1, corner A of the rectangular plate has a velocity v A = 2.8 m/s and the plate has a clockwise angular velocity ω = 12 rad/s. Determine: (a) the distance between A and the instantaneous centre C (of zero velocity); (b) the distance between B and the instantaneous centre C; and (c) the magnitude of the corresponding velocity of point B. 2. (Problem 5/127 from Meriam and Kraige - Dynamics) The 9 m steel beam is being hoisted from its horizontal position by the two cables attached at A and B. If the initial angular accelerations of the hoisting drums are α 1 = 0.5 rad/s 2 and α 2 = 0.2 rad/s 2 in the directions shown in Figure 2, determine: (a) the corresponding angular acceleration α of the beam,; (b) the acceleration of C; and (c) and the distance b from B to a point P on the beam centerline which has no acceleration. 3. (Problem 5/128 from Meriam and Kraige - Dynamics) A car has a forward acceleration a = 4 m/s 2 without slipping its 600 mm diameter tyres. When a point P on the tire in the position shown in Figure 3 have zero horizontal component of acceleration, determine: Figure 1: Rectangle plate 1
Figure 2: Hoisted beam Figure 3: Tyre (a) the tangential component of relative acceleration (a P/O ) t ; (b) the normal component of relative acceleration (a P/O ) n ; and (c) the velocity v of the car. 4. (Problem 6/1, 6/2 from Meriam and Kraige - Dynamics) (a) Accelerating frame (b) Overhead monorail system Figure 4: Accelerating objects (a) The uniform 30 kg bar OB shown in Figure 4(a) is secured in the vertical position to the accelerating frame by the hinge at O and the roller at A. If the horizontal acceleration of the frame is a = 20 m/s 2, compute the force F A on the roller; (b) and the horizontal component of the force supported by the pin at O. Page 2 of 8
(c) A passenger car of an overhead monorail system is driven by one of its two small wheels A or B (see Figure 4(b)). Select the one for which the car can be given the greater acceleration without slipping the driving wheel and compute the maximum acceleration if the effective coefficient of friction is limited to 0.25 between the wheels and the rail. Neglect the small mass of the wheels. 5. (Problem 6/35, 6/36 from Meriam and Kraige - Dynamics) (a) Hinged plate (b) Automotive dynamometer Figure 5: Hinged plate and automotive dynamometer (a) The 20 kg uniform steel plate is freely hinged about the z-axis as shown in Figure 5(a). Calculate the tangential acceleration of the mass centre of the plate when it is released from rest in the horizontal y-z plane; and (b) calculate the force supported by each of the bearings at A and B at this instant. (c) The automotive dynamometer is able to simulate road conditions for an acceleration of 0.6g for the loaded pickup truck with a total mass of 2.8 Mg (see Figure 5(b)). Calculate the required moment of inertia of the dynamometer drum about its centre O assuming that the drum turns freely during the acceleration phase of the test. 6. (Problem 6/76, 6/86 from Meriam and Kraige - Dynamics) (a) The 30 kg solid circular disk is initially at rest on the horizontal surface when a 12 N force P, constant in magnitude and direction, is applied to the cord wrapped securely around its periphery (see Figure 6(a)). Friction between the disc and the surface is negligible. Calculate the linear velocity v of the centre of the disk after it has moved 1.2 metres from rest; and (b) find the angular velocity ω of the disk after the 12 N force has been applied for 2 seconds. Page 3 of 8
(a) Circular disc (b) Hoisted beam Figure 6: Circular disc and hoisted beam (c) The 3.6 m steel beam shown in Figure 6(b) has a mass of 125 kg and is hoisted from rest where the tension in each of the cables is 613 N. If the hoisting drums are given initial angular accelerations α 1 = 4 rad/s 2 and α 2 = 6 rad/s 2, calculate the corresponding tensions T A and T B in the cables. The beam may be treated as a slender bar. Part B 7. (Problem 5/95, 5/97, 5/98 from Meriam and Kraige - Dynamics) (a) The bar AB shown in Figure 7(a) has a counterclockwise angular velocity of 6 rad/s. Construct the velocity vectors for points A and G of the bar and specify their magnitudes if the instantaneous centre of zero velocity for the bar is (a) at C 1, (b) at C 2, and (c) at C 3. (b) At a certain instant vertex B of the right-triangular plate has a velocity of 200 mm/s in the direction shown in Figure 7(b). If the instantaneous centre of zero velocity for the plate is 40 mm from point B and if the angular velocity of the plate is clockwise, determine the velocity of point D. (c) A car mechanic walks two wheel/tyre units across a horizontal floor as shown in Figure 7(c). He walks with constant speed v and keeps the tires in the configuration shown with the same position relative to his body. If there is no slipping at any interface, determine (a) the angular velocity of the lower tyre, (b) the angular velocity of the upper tyre, and (c) the velocities of points A, B, C, and D. The radius of both tires is r. Page 4 of 8
(a) Bar (b) Right-triangular plate (c) Rolling wheels Figure 7: Instantaneous centre problems 8. (Problem 5/129, 5/135 from Meriam and Kraige - Dynamics) (a) The centre O of the disk has the velocity and acceleration shown in Figure 8(a). If the disc rolls without slipping on the horizontal surface, determine the velocity of A and the acceleration of B for the instant represented. (b) If the velocity of point A is 3 m/s to the right and is constant for an interval including the position shown in Figure 8(b), determine the tangential acceleration of point B along its path and the angular acceleration of the bar. 9. (Problem 5/145 from Meriam and Kraige - Dynamics) If OA in the linkage shown in Figure 9 has a constant counterclockwise angular velocity ω 0 = 10 rad/s, calculate the angular acceleration of link AB for the position where the coordinates of A are x = 60 mm and y = 80 mm. Link BC is vertical for this position. Solve by vector algebra. Given that v BC = 5.83k rad/s and v AB = 2.5k rad/s.) 10. (Problem 5/146 from Meriam and Kraige - Dynamics) The revolving crank ED and connecting link CD cause the rigid frame ABO to oscillate about O (see Figure 10). For the instant represented ED and CD are both perpendicular to F O, and the crank ED has an angular velocity of 0.4 rad/s and an angular acceleration of 0.06 rad/s 2, both counterclockwise. For this instant determine the acceleration of point A with respect to point B. Page 5 of 8
(a) Disc (b) Bar Figure 8: Disc and bar Figure 9: Linkage 11. (Problem 6/18 from Meriam and Kraige - Dynamics) The device shown in Figure 11 oscillates horizontally according to x = b sin ωt, where b and ω are constants. Determine and plot the force T in the light link at A as a function of the time t. The mass of the uniform slender rod AP is m. 12. (Problem 6/29 from Meriam and Kraige - Dynamics) Determine the maximum mass m of the cylinder for which the loaded 2000 kg coal car will not overturn about the rear wheels B (see Figure 12). Neglect the mass of all pulleys and wheels. (Note that the tension in the cable at C is not 2mg.) 13. (Problem 6/54 from Meriam and Kraige - Dynamics) A device for impact testing consists of a 34 kg pendulum with mass centre at G and with radius of gyration about O of 620 mm (see Figure 13). The distance b for the pendulum is selected so that the force on the bearing at O has the least possible value during impact with the specimen at the Page 6 of 8
Figure 10: Crank Figure 11: Oscillating device Figure 12: Coal car Figure 13: Impact testing machine Page 7 of 8
Figure 14: Coal car Figure 15: Dump truck bottom of the swing. Determine b and calculate the magnitude of the total force R on the bearing O an instant after release from rest at θ = 60. 14. (Problem 6/55 from Meriam and Kraige - Dynamics) The 12 kg cylinder supported by the bearing brackets at A and B has a moment of inertia about the vertical z 0 -axis through its mass centre G equal to 0.080 kgm 2 (see Figure 14). The disk and brackets have a moment of inertia about the vertical z-axis of rotation equal to 0.60 kgm 2. If a torque M = 16 Nm is applied to the disk through its shaft with the disc initially at rest, calculate the horizontal x-components of force supported by the bearings at A and B. 15. (Problem 6/105 from Meriam and Kraige - Dynamics) The hydraulic cylinder BC of the dump truck is broken and is disconnected (see Figure 15). The driver (who has passed a course in dynamics) decides to calculate the minimum acceleration a of the truck required to tilt the dump about its pivot at A. He then proceeds to calculate the initial angular acceleration α of the dump if the truck is given an acceleration of 1.2a. What are his correct answers for a and α, and would he be able to carry out the experiment? The dump container may be modelled as a homogeneous and solid rectangular block with mass centre at G. Page 8 of 8