mech 220 mechanics 2
MECH 220 MECHANICS 2 FINAL EXAMINATION Wednesday December 11, 2002 Examiner:
Prof. I. Sharf
Associate Examiner:
Prof. A. Misra
14:00 to 17:00
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY: • THIS EXAM IS CLOSED BOOK. ONE 8.5” x 11” CRIBSHEET, HANDWRITTEN ON BOTH SIDES, IS PERMITTED. • PLEASE ANSWER ALL SIX (6) QUESTIONS. • QUESTIONS MUST BE ANSWERED DIRECTLY ON THE EXAMINATION PAPER, IN THE SPACE PROVIDED BELOW OR BESIDE THE QUESTIONS. • FACULTY STANDARD CALCULATOR PERMITTED. STUDENT NAME: STUDENT NUMBER:
_________________________
FOR EXAMINERS ONLY 1.
_________________________ 2. 3. 4.
5.
6.
Total:
PROBLEM 1 [20] The track shown is contained in a vertical plane. An object of mass m is given an initial velocity u up the incline at point A. An instant after it passes point B, the normal force of contact between it and the track drops to one half the value it had just as it approached B. The coefficient of sliding friction between the object and the track is 0.3. a. Find the velocity of the object at point B. b. Find the velocity of the object at point A. (Note, if you have not been able to solve part (a), assume that the velocity at B was 1 m/s).
Mech 220/02
1
FINAL/02
PROBLEM 2
[20]
At the instant θ = 60°, the pinned rod has an angular velocity of 3 rad/s and an angular acceleration of 2 rad/s2. At this same instant, the collar C is traveling outward along the rod such that when x=0.2 m the velocity is 2 m/s and the acceleration is 3 m/s2, both measured relative to the rod. Determine: a. Velocity of the collar at this instant; b. Acceleration of the collar at this instant; c. Normal acceleration of the collar.
Mech 220/02
2
FINAL/02
PROBLEM 3
[20]
The disk has a mass of 5 kg, a center of gravity at G, and a radius of gyration about point O of kO = 80 mm. A cord is placed over the rim of the disk, and the attached blocks A and B are released from rest when the disk is in the position shown. If the cord does not slip on the rim, determine the angular velocity of the disk when it has rotated 180°. Blocks A and B have a mass of 4 kg and 10 kg respectively.
Mech 220/02
3
FINAL/02
PROBLEM 4 [20] The 4-kg rectangular plate lies in the x-y plane of the body-fixed coordinate system. If the plate is rotating about the fixed point O with angular velocity ω = 4i − 2 j rad/s: a. Determine the plate’s angular momentum about O; b. Determine the plate’s kinetic energy.
Note:
Mech 220/02
4
FINAL/02
[20] PROBLEM 5 The horizontal velocity of the landing airplane is 50 m/s, its vertical velocity (rate of descent) is 2 m/s, and its angular velocity is zero. The mass of the airplane is 12 Mg and the mass moment of inertia about its center of mass is 1 x 105 kg-m2. When the rear wheels touch the runway, they remain in contact with it. Neglecting the horizontal force exerted on the wheels by the runway, determine the airplane’s angular velocity just after it touches down.
Mech 220/02
5
FINAL/02
[20] PROBLEM 6 Arm BC has a mass of 12 kg, and its moments and products of inertia with respect to center of mass frame parallel to the coordinate system shown are Ixx = 0.03 kg-m2, Iyy = Izz = 2.92 kg-m2, Ixy = Ixz = Iyz= 0. At the instant shown, arm AB is rotating in the horizontal plane with a constant angular velocity of 1 rad/s in the counterclockwise direction viewed from above. Relative to arm AB, arm BC is rotating about the z-axis with a constant angular velocity of 2 rad/s. Determine the force and couple vectors exerted on arm BC at B.
Mech 220/02
6
FINAL/02
Prof. I. Sharf
Associate Examiner:
Prof. A. Misra
14:00 to 17:00
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY: • THIS EXAM IS CLOSED BOOK. ONE 8.5” x 11” CRIBSHEET, HANDWRITTEN ON BOTH SIDES, IS PERMITTED. • PLEASE ANSWER ALL SIX (6) QUESTIONS. • QUESTIONS MUST BE ANSWERED DIRECTLY ON THE EXAMINATION PAPER, IN THE SPACE PROVIDED BELOW OR BESIDE THE QUESTIONS. • FACULTY STANDARD CALCULATOR PERMITTED. STUDENT NAME: STUDENT NUMBER:
_________________________
FOR EXAMINERS ONLY 1.
_________________________ 2. 3. 4.
5.
6.
Total:
PROBLEM 1 [20] The track shown is contained in a vertical plane. An object of mass m is given an initial velocity u up the incline at point A. An instant after it passes point B, the normal force of contact between it and the track drops to one half the value it had just as it approached B. The coefficient of sliding friction between the object and the track is 0.3. a. Find the velocity of the object at point B. b. Find the velocity of the object at point A. (Note, if you have not been able to solve part (a), assume that the velocity at B was 1 m/s).
Mech 220/02
1
FINAL/02
PROBLEM 2
[20]
At the instant θ = 60°, the pinned rod has an angular velocity of 3 rad/s and an angular acceleration of 2 rad/s2. At this same instant, the collar C is traveling outward along the rod such that when x=0.2 m the velocity is 2 m/s and the acceleration is 3 m/s2, both measured relative to the rod. Determine: a. Velocity of the collar at this instant; b. Acceleration of the collar at this instant; c. Normal acceleration of the collar.
Mech 220/02
2
FINAL/02
PROBLEM 3
[20]
The disk has a mass of 5 kg, a center of gravity at G, and a radius of gyration about point O of kO = 80 mm. A cord is placed over the rim of the disk, and the attached blocks A and B are released from rest when the disk is in the position shown. If the cord does not slip on the rim, determine the angular velocity of the disk when it has rotated 180°. Blocks A and B have a mass of 4 kg and 10 kg respectively.
Mech 220/02
3
FINAL/02
PROBLEM 4 [20] The 4-kg rectangular plate lies in the x-y plane of the body-fixed coordinate system. If the plate is rotating about the fixed point O with angular velocity ω = 4i − 2 j rad/s: a. Determine the plate’s angular momentum about O; b. Determine the plate’s kinetic energy.
Note:
Mech 220/02
4
FINAL/02
[20] PROBLEM 5 The horizontal velocity of the landing airplane is 50 m/s, its vertical velocity (rate of descent) is 2 m/s, and its angular velocity is zero. The mass of the airplane is 12 Mg and the mass moment of inertia about its center of mass is 1 x 105 kg-m2. When the rear wheels touch the runway, they remain in contact with it. Neglecting the horizontal force exerted on the wheels by the runway, determine the airplane’s angular velocity just after it touches down.
Mech 220/02
5
FINAL/02
[20] PROBLEM 6 Arm BC has a mass of 12 kg, and its moments and products of inertia with respect to center of mass frame parallel to the coordinate system shown are Ixx = 0.03 kg-m2, Iyy = Izz = 2.92 kg-m2, Ixy = Ixz = Iyz= 0. At the instant shown, arm AB is rotating in the horizontal plane with a constant angular velocity of 1 rad/s in the counterclockwise direction viewed from above. Relative to arm AB, arm BC is rotating about the z-axis with a constant angular velocity of 2 rad/s. Determine the force and couple vectors exerted on arm BC at B.
Mech 220/02
6
FINAL/02
