The University of Melbourne 436-291 Engineering Mechanics Tutorial Twelve General Plane Motion, Work and Energy Part A (Introductory) 1. (Problem 6/78 from Meriam and Kraige - Dynamics) Above the earth s atmosphere at an altitude of 400 km where the acceleration due to gravity is 8.69 m/s 2, a certain rocket has a total remaining mass of 300 kg and is directed 30 from the vertical. The rocket has a centroidal radius of gyration of 1.5 m. If the thrust T from the rocket motor is 4 kn and if the rocket nozzle is tilted through an angle of 1 as shown in Figure 1, calculate: (a) the angular acceleration α of the rocket; (b) x-component of the acceleration of its mass centre G; and (c) y-component of the acceleration of its mass centre G. 2. (Problem 6/79, 6/81 from Meriam and Kraige - Dynamics) (a) The solid homogeneous cylinder is released from rest on the ramp (see Figure 2(a). If θ = 40, μ s = 0.30, and μ k = 0.20, determine the acceleration of the mass centre G; and (b) the friction force exerted by the ramp on the cylinder. (c) What should be the radius r 0 of the circular groove (see Figure 2(b)) in order that there be no friction force acting between the wheel and the horizontal surface regardless of the magnitude of the force P applied to the cord? The centroidal radius of gyration of the wheel is k. 3. (Problem 6/83, 6/85 from Meriam and Kraige - Dynamics) Figure 1: Rocket 1
(a) Cylinder (b) Wheel Figure 2: Cylinder and wheel (a) Square panel (b) Circuilar disc Figure 3: Square panel and circular disc (a) The uniform 12 kg square panel shown in Figure 3(a) is suspended from point C by the two wires at A and B. If the wire at B suddenly breaks, show that the angular acceleration of the square panel is α = 3g 4b at an instant after the break occurs. (b) Then, calculate the tension T in the wire at A an instant after the break occurs. (c) The circular disk of mass m and radius r shown in Figure 3(b) is rolling through the bottom of the circular path of radius R. If the disk has an angular velocity ω, determine the force N exerted by the path on the disk. 4. (Problem 6/115, 6/118 from Meriam and Kraige - Dynamics) (a) Slender rod (b) Battering ram Figure 4: Slender rod and battering ram Page 2 of 7
Figure 5: Square frame Figure 6: Hinged bars (a) The slender rod of mass m and length l has a particle (negligible radius, mass 2 m) attached to its end (see Figure 4(a)). Show that the moment of inertia for the rod about O is I O = 61 48 ml2. (b) If the body is released from rest when in the position shown, determine its angular velocity as it passes the vertical position. (c) The log is suspended by the two parallel 5 m cables and used as a battering ram (see Figure 4(b)). At what angle θ should the log be released from rest in order to strike the object to be smashed with a velocity of 4 m/s? 5. (Problem 6/122 from Meriam and Kraige - Dynamics) The square frame shown in Figure 5 is made from four slender rods, each of mass m and length b. The frame is rotating in its plane with an angular velocity ω. Determine: (a) the kinetic energy of rotation; (b) the kinetic energy of translation; and (c) the linear velocity v of the centre C which will make the kinetic energy of translation equal to the kinetic energy of rotation. 6. (Problem 6/124 from Meriam and Kraige - Dynamics) Page 3 of 7
Figure 7: Bowling Figure 8: Car crash Each of the hinged bars has a mass ρ per unit length, and the assembly is suspended at O in the vertical plane (see Figure 6). The bars are released from rest with θ essentially zero, and reach the final position when A and B and C and D come together. Determine: (a) the change of potential energy ΔV g of the whole assembly; (b) the change of potential energy ΔT of the whole assembly; and (c) the angular velocity ω common to all bars at the final position. Part B 7. (Problem 6/106 from Meriam and Kraige - Dynamics) A 6.4 kg bowling ball with a circumference of 690 mm has a radius of gyration of 83 mm (see Figure 7). If the ball is released with a velocity of 6 m/s but with no angular velocity as it touches the alley floor, compute the distance travelled by the ball before it begins to roll without slipping. The coefficient of friction between the ball and the floor is 0.20. 8. (Problem 6/107 from Meriam and Kraige - Dynamics) In a study of head injury against the instrument panel of a car during sudden or crash stops where lap belts without shoulder straps or airbags are used, the segmented human model shown in Figure 8 is analysed. The hip joint O is assumed to remain fixed relative to the car, and the torso above the hip is treated as a rigid body of mass m freely pivoted at O. The centre of mass of the torso is at G with the initial position of OG taken as vertical. The radius of gyration of the torso about O is k O. If the car is brought to a sudden stop with a constant deceleration a, determine the velocity v relative to the car with which the model s head strikes the instrument panel. Substitute the values m = 50 kg, r = 450 mm, r = 800 mm, k O = 550 mm, θ = 45, and a = 10g and compute v. 9. (Problem 6/111 from Meriam and Kraige - Dynamics) The 0.6 kg connecting rod AB of a certain internal-combustion engine has a mass centre at G and a radius of gyration about G of 28 mm (see Figure 9). The piston and piston pin A have a combined mass of 0.82 kg. The engine is running at a constant speed of 3000 rev/min, so that the angular Page 4 of 7
Figure 9: Car crash velocity of the crank is 3000(2π)/60 = 100π rad/s. Neglect the weights of the components and the force exerted by the gas in the cylinder compared with the dynamic forces generated and calculate the magnitude of the force on the piston pin A for the crank angle θ = 90. (Suggestion: Use the alternative moment relation, Eq. 6/3 ( M P = I P α + ρ ma p ), with B as the moment centre.) 10. (Problem 6/114 from Meriam and Kraige - Dynamics) The Ferris wheel at an amusement park has an even number n of gondolas, each freely pivoted at its point of support on the wheel periphery (see Figure 10). Each gondola has a loaded mass m, a radius of gyration k about its point of support A, and a mass centre a distance h from A. The wheel structure has a moment of inertia I O about its bearing at O. Determine an expression for the tangential force F which must be transmitted to the wheel periphery at C in order to give the wheel an initial angular acceleration α starting from rest. Suggestion: Analyse the gondolas in pairs A and B. Be careful not to assume that the initial angular acceleration of the gondolas is the same as that of the wheel. (Note: An American engineer named George Washington Gale Ferris, Jr., created a giant amusement-wheel ride for the World s Columbian Exposition in Chicago in 1893. The wheel was 250 ft in diameter with 36 gondolas, each of which carried up to 60 passengers. Fully loaded, the wheel and gondolas had a mass of 1200 tons. The ride was powered by a 1000 hp steam engine.) 11. (Problem 6/137 from Meriam and Kraige - Dynamics) The semicircular disk of mass m = 2 kg is mounted in the light hoop of radius r = 150 mm and released from rest in position (a) (see Figure 11). Determine the angular velocity ω of the hoop and the normal force N under the hoop as it passes position (b) after rotating through 180. The hoop rolls without slipping. 12. (Problem 6/138 from Meriam and Kraige - Dynamics) The electric motor shown in Figure 12 is delivering 4 kw at 1725 rev/min to a pump which it drives. Calculate the angle δ through which the motor deflects under load if the stiffness of each of its four spring mounts is 15 kn/m. In what direction does the motor shaft turn? 13. (Problem 6/144 from Meriam and Kraige - Dynamics) The body shown in Figure 13 is constructed of uniform slender rod and consists of a ring of radius r attached to a straight section of length 2r. The body pivots freely about a ball-and-socket joint Page 5 of 7
Figure 10: Ferris wheel Figure 11: Light hoop with semicircular disc Figure 12: Electric motor Figure 13: Pivoted body Page 6 of 7
Figure 14: Bus Figure 15: Open square frame at O. If the body is at rest in the vertical position shown and is given a slight nudge, compute its angular velocity ω after a 90 rotation about (a) axis A-A (b) axis B-B. 14. (Problem 6/146 from Meriam and Kraige - Dynamics) Motive power for the experimental 10 Mg bus comes from the energy stored in a rotating flywheel which it carries (see Figure 14). The flywheel has a mass of 1500 kg and a radius of gyration of 500 mm and is brought up to a maximum speed of 4000 rev/min. If the bus starts from rest and acquires a speed of 72 km/h at the top of a hill 20 m above the starting position, compute the reduced speed N of the flywheel. Assume that 10 percent of the energy taken from the flywheel is lost. Neglect the rotational energy of the wheels of the bus. The 10 Mg mass includes the flywheel. 15. (Problem 6/154 from Meriam and Kraige - Dynamics) The open square frame is constructed of four identical slender rods, each of length b. The small wheels roll without friction in the slots of the vertical surface. If the frame is released from rest in the position shown in Figure 15, determine the speed of corner A (a) after A has dropped a distance b; and (b) after A has dropped a distance 2b. Page 7 of 7