15-55

ME 3112 Mechanics of Machines

Problem Set 1: Particle DynamicsIn this problem set, Problems 1 to 4 will be discussed during the tutorial session. Problems 5 to 7 are given as extras for your own practice.

Problem 1A baseball pitching machine throws baseballs with a horizontal velocity v0. Knowing that height h varies between 788 mm and 1068 mm, determine (a) the range of values of v0, (b) the values of ( corresponding to h = 788 mm and h = 1068 mm.Answer: (a) 32.0 m/s; 41.1 m/s. (b) 4.05; 6.67

Problem 2As the truck shown begins to back up with a constant acceleration of 1.2 m/s2, the outer section B of its boom starts to retract with a constant acceleration of 0.5 m/s2 relative to the truck. Determine (a) the acceleration of section B, (b) the velocity of section B when t=2 s.

Answer: (a) (0.879 i 0.383 j) m/s2. (b) (1.758 i 0.766 j) m/s

Problem 3A turntable A is built into a stage for use in a theatrical production. It is observed during a rehearsal that a trunk B starts to slide on the turntable 12 s after the turntable begins to rotate. Knowing that the trunk undergoes a constant tangential acceleration of 0.25 m/s2, determine the coefficient of static friction between the trunk and the turntable.Answer: 0.383

Problem 4To unload a bound stack of plywood from a truck, the driver first tilts the bed of the truck and then accelerates from rest. Knowing the coefficients of friction between the bottom sheet of plywood and the bed are s = 0.40 and k = 0.30, determine (a) the smallest acceleration of the truck which will cause the stack of plywood to slide, (b) the acceleration of the truck which causes corner A of the stack to reach the end of the bed in 0.9 s.

Answer: (a) 0.31 m/s2 (b) 4.17 m/s2

Problem 5 (Extra)A telemetry system is used to quantify kinematic values of a ski jumper immediately before she leaves the ramp. According to the system r = 150m, dr/dt = -31.5 m/s, d2r/dt2 = -3 m/s2, = 25, d/dt = 0.07 rad/s, d2/dt2 = 0.06 rad/s2. Determine (a) the velocity of the skier immediately before she leaves the jump, (b) the acceleration of the skier at this instant, (c) the distance of the jump d neglecting lift and air resistance.

Problem 6 (Extra)Block A has a mass of 40 kg, and block B has a mass of 8 kg. The coefficients of friction between all surfaces of contact are s = 0.20 and k = 0.15. If P = 40 N, determine (a) the acceleration of block B, (b) the tension in the cord.

Problem 7 (Extra)A 3-kg block is at rest relative to a parabolic dish which rotates at a constant rate about a vertical axis. Knowing that the coefficient of static friction is 0.5 and that r = 2 m, determine the maximum allowable velocity of the block.

ME 3112 Mechanics of Machines

Problem Set 2: Kinematics of rigid bodies

In this problem set, Problems 1 to 4 will be discussed during the tutorial session. Problems 5 to 7 are given as extras for your own practice.

Problem 1Knowing that the disk has a constant angular velocity of 15 rad/s clockwise, determine the angular velocity of bar BD and the velocity of collar D when (a) =0(, (b) =90(, (c) =180(.

Answer: (a) 4.38 rad/s; 0.31 m/s. (b) 0 rad/s; 1.065 m/s. (c) 4.38 rad/s; 0.31 m/s.

Problem 2Collar D slides on a fixed vertical rod. Knowing that the disk has a constant angular velocity of 15 rad/s clockwise, determine the angular acceleration of bar BD and the acceleration of collar D when (a) =0(, (b) =90(, (c) =180(.

Answer: (a)5.55 rad/s2; 10.93m/s2. (b) 75.9 rad/s2; 10.77m/s2. (c) 5.55 rad/s2; 21.02m/s2.

Problem 3The motion of nozzle D is controlled by arm AB. At the instant shown the arm is rotating counterclockwise at the constant rate =2.4 rad/s and portion BC is being extended at the constant rate u=250 mm/s with respect to the arm. For each of the arrangements shown, determine the acceleration of nozzle D.

Answer: (a) 1.702 m/s2. (b) 2.55 m/s2

Problem 4The crane shown rotates at the constant rate =0.25 rad/s; simultaneously, the telescoping boom is being lowered at the constant rate =0.40 rad/s. Knowing that at the instant shown the length of the boom is 6m and is increasing at the constant rate u=0.45m/s, determine the velocity and acceleration of point B.Answer: v=(1.299i-1.853j+1.590k) m/s, a=(0.795i-0.792j-0.976k)m/s2

Problem 5 (Extra)Arm AB has a constant angular velocity of 16 rad/s counter-clockwise. At the instant when = 90, determine the acceleration (a) of collar D, (b) of the midpoint G of bar BD.

Problem 6 (Extra)Pin P is attached to the wheel shown and slides in a slot cut in bar BD. The wheel rolls to the right without slipping with a constant angular velocity of 20 rad/s. Knowing that x = 480 mm when = 0, determine (a) the angular acceleration of the bar, (b) the relative acceleration of pin P with respect to the bar when = 0.

Problem 7 (Extra)A disk of radius 120 mm rotates at the constant rate 2 = 5 rad/s with respect to the arm AB, which itself rotates at the constant rate 1 = 3 rad/s. For the position shown, determine the velocity and acceleration of point C.

ME 3112 Mechanics of Machines

Problem Set 3: Kinetics and Newtons second lawIn this problem set, Problems 1 to 3 will be discussed during the tutorial session. Problems 4 to 6 are given as extras for your own practice.

Problem 1A 4 kg uniform slender rod AB is held in position by two ropes and the link CA which has a negligible weight. After rope BD is cut the assembly rotates in a vertical plane under the combined effect of gravity and a 6 Nm couple M applied to link CA as shown. Determine, immediately after rope BD has been cut, (a) the acceleration of rod AB, (b) the tension in rope EB.

Answer: (a) 1.572 m/s2. (b) 20.8 N.

Problem 2The collar B of negligible mass can slide freely on the 4 kg uniform rod CD. Knowing that in the position shown crank AB rotates with an angular velocity of 5 rad/s and an angular acceleration of 60 rad/s2, both clockwise, determine the force exerted on rod CD by collar B.

Answer: 15.01 N

Problem 3A thin ring of radius r=20.3 cm is attached by a collar at point A to a vertical shaft which rotates with a constant angular velocity . Determine (a) the constant angle that the plane of the ring forms with the vertical when =10 rad/s, (b) the largest value of for which the ring will remain vertical (=0).

Answer: (a) 71.2(. (b)5.67 rad/s.

Problem 4 (Extra)A 300-mm radius cylinder of mass 8 kg rests on a 3-kg carriage. The system is at rest when a force P of magnitude 20 N is applied. Knowing that the cylinder rolls without sliding on the carriage and neglecting the mass of the wheels of the carriage, determine (a) the acceleration of the carriage, (b) the acceleration of point A, (c) the distance the cylinder has rolled with respect to the carriage after 0.5 s.

Problem 5 (Extra)The 250-mm uniform rod BD, of mass 5 kg, is connected as shown to disk A and to a collar of negligible mass, that may slide freely along a vertical rod. Knowing that disk A rotates counterclockwise at a constant rate of 500 rpm, determine the reactions at D when = 0.

Problem 6 (Extra)The 950-g gear A is constrained to roll on the fixed gear B, but is free to rotate about axle AD. Axle AD, of length 400 mm and negligible mass, is connected by a clevis to the vertical shaft DE which rotates as shown with a constant angular velocity 1. Assuming that gear A can be approximated by a thin disk of radius 80 mm, determine the largest allowable value of 1 if gear A is not to lose contact with gear B.