mech 220 mechanics 2 final examination
MECH 220 MECHANICS 2 FINAL EXAMINATION Friday December 14, 2004 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 OR TYPED 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
[15]
The 20-kg cylinder is released at the position shown and falls onto the linear spring with stiffness k = 3000N/m. If friction in the guides can be neglected, determine how far down the cylinder moves after contacting the spring.
Mech 220/04
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Mech 220/04
2
FINAL/04
PROBLEM 2
[20]
The drum has a mass of 70 kg, a radius of 300 mm, and radius of gyration kO = 125 mm. If the coefficients of static and kinetic friction at A are µ s = 0.4 and µ k = 0.3 respectively, and
θ = 30o , determine a) the drum’s angular velocity 3 s after it is released from rest; b) the drum’s kinetic energy 3 s after it is released from rest.
Mech 220/04
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Mech 220/04
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[15]
PROBLEM 3
The crane has a boom of length OP = 24 m and is revolving about the vertical axis at the constant rate of 2 rev/min in the direction shown. Simultaneously, the boom is being lowered at the constant rate β& . When the crane passes the position β = 30o , the tip of the crane, i.e., point P, is observed to be descending at a rate of 1.5 m/s. a) What is the rate β& in rad/s at this instant? b) What is the angular acceleration of the boom at this instant? If you did not answer a), choose a reasonable value of β& to answer b).
Mech 220/04
5
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Mech 220/04
6
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[25]
PROBLEM 4
The system in the figure below is released from rest in the position shown. Disk m1 and pulley m2 are rigidly mounted to massless rod AB, which is mounted horizontally in two frictionless ball-and-socket joints at points A and B. The disk is mounted off-center a distance d from its central axis, with the centroid initially at its vertical peak. A mass m3 hangs from the pulley by a massless string. The radii of the disk and pulley, respectively, are r1 and r2. Parameter values are: m1 = 3 kg, m2 = 0.5 kg, m3 = 2 kg, r1 = 10 cm, r2 = 4 cm, d = 6 cm, and L1 = L2 = L3 = 15 cm. Using the x-y-z axes fixed to rod AB as shown, calculate: a) The moment of inertia Izz about the axis of rotation and the product of inertia Iyz of the disk-pulley-mass system; b) The tension in the string at the instant of release; c) The vertical force reaction at support B at the instant of release.
Mech 220/04
7
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Mech 220/04
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[25]
PROBLEM 5
The combined mass of the motorcycle and rider is 160 kg. The rear wheel exerts a 400-N horizontal force on the road, and you can neglect the horizontal force exerted on the road by the front wheel. The radius of the wheels is 300 mm, the moment of inertia of the rear wheel about its axle is I = 0.7 kg-m2 and center of mass of the motorcycle is 660 mm above the ground. Assuming the wheels do not slip, determine: a) The motorcycle’s acceleration; b) The normal forces exerted on the road by the rear and front wheels; c) The torque exerted by the engine on the rear wheel; d) If the torque on the rear wheel remains constant, what horizontal distance must the motorcycle travel to reach the velocity of 25 m/s starting from rest?
Mech 220/04
9
FINAL/04
Mech 220/04
10
FINAL/04
[20]
PROBLEM 6
You are asked to evaluate two types of locomotion for designing a planetary exploration robot: hop-type and wheel-type. As the first basis for comparison, you decide to use the distance covered in a certain time. Carry out basic kinematics/dynamics analysis to determine the information below in order to compare the two types of locomotion on flat ground. State any relevant assumptions and use the mass m = 1kg for both hop-type or wheel-type robot, coefficient of friction µ = 0.8 and the gravitational environment as 10% of that on Earth. a) Calculate the duration of one hop and the distance covered in one hop for the hopping robot. Assume the “pushing” force generated by the hopping robot is an impulsive force giving the robot a pushing impulse of 0.2 N-sec and the “driving” force is the maximum friction force generated by the surface. b) Calculate the distance covered by wheel-type robot in the duration of one hop determined in (a) above. Here too assume the maximum driving force that can be generated by the surface. c) How do the distances calculated in a) and b) above compare? Which assumption in your analysis would you say is most questionable? What would be your next step in this comparison?
Mech 220/04
11
FINAL/04
Mech 220/04
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Appendix
Mech 220/04
13
FINAL/04
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 OR TYPED 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
[15]
The 20-kg cylinder is released at the position shown and falls onto the linear spring with stiffness k = 3000N/m. If friction in the guides can be neglected, determine how far down the cylinder moves after contacting the spring.
Mech 220/04
1
FINAL/04
Mech 220/04
2
FINAL/04
PROBLEM 2
[20]
The drum has a mass of 70 kg, a radius of 300 mm, and radius of gyration kO = 125 mm. If the coefficients of static and kinetic friction at A are µ s = 0.4 and µ k = 0.3 respectively, and
θ = 30o , determine a) the drum’s angular velocity 3 s after it is released from rest; b) the drum’s kinetic energy 3 s after it is released from rest.
Mech 220/04
3
FINAL/04
Mech 220/04
4
FINAL/04
[15]
PROBLEM 3
The crane has a boom of length OP = 24 m and is revolving about the vertical axis at the constant rate of 2 rev/min in the direction shown. Simultaneously, the boom is being lowered at the constant rate β& . When the crane passes the position β = 30o , the tip of the crane, i.e., point P, is observed to be descending at a rate of 1.5 m/s. a) What is the rate β& in rad/s at this instant? b) What is the angular acceleration of the boom at this instant? If you did not answer a), choose a reasonable value of β& to answer b).
Mech 220/04
5
FINAL/04
Mech 220/04
6
FINAL/04
[25]
PROBLEM 4
The system in the figure below is released from rest in the position shown. Disk m1 and pulley m2 are rigidly mounted to massless rod AB, which is mounted horizontally in two frictionless ball-and-socket joints at points A and B. The disk is mounted off-center a distance d from its central axis, with the centroid initially at its vertical peak. A mass m3 hangs from the pulley by a massless string. The radii of the disk and pulley, respectively, are r1 and r2. Parameter values are: m1 = 3 kg, m2 = 0.5 kg, m3 = 2 kg, r1 = 10 cm, r2 = 4 cm, d = 6 cm, and L1 = L2 = L3 = 15 cm. Using the x-y-z axes fixed to rod AB as shown, calculate: a) The moment of inertia Izz about the axis of rotation and the product of inertia Iyz of the disk-pulley-mass system; b) The tension in the string at the instant of release; c) The vertical force reaction at support B at the instant of release.
Mech 220/04
7
FINAL/04
Mech 220/04
8
FINAL/04
[25]
PROBLEM 5
The combined mass of the motorcycle and rider is 160 kg. The rear wheel exerts a 400-N horizontal force on the road, and you can neglect the horizontal force exerted on the road by the front wheel. The radius of the wheels is 300 mm, the moment of inertia of the rear wheel about its axle is I = 0.7 kg-m2 and center of mass of the motorcycle is 660 mm above the ground. Assuming the wheels do not slip, determine: a) The motorcycle’s acceleration; b) The normal forces exerted on the road by the rear and front wheels; c) The torque exerted by the engine on the rear wheel; d) If the torque on the rear wheel remains constant, what horizontal distance must the motorcycle travel to reach the velocity of 25 m/s starting from rest?
Mech 220/04
9
FINAL/04
Mech 220/04
10
FINAL/04
[20]
PROBLEM 6
You are asked to evaluate two types of locomotion for designing a planetary exploration robot: hop-type and wheel-type. As the first basis for comparison, you decide to use the distance covered in a certain time. Carry out basic kinematics/dynamics analysis to determine the information below in order to compare the two types of locomotion on flat ground. State any relevant assumptions and use the mass m = 1kg for both hop-type or wheel-type robot, coefficient of friction µ = 0.8 and the gravitational environment as 10% of that on Earth. a) Calculate the duration of one hop and the distance covered in one hop for the hopping robot. Assume the “pushing” force generated by the hopping robot is an impulsive force giving the robot a pushing impulse of 0.2 N-sec and the “driving” force is the maximum friction force generated by the surface. b) Calculate the distance covered by wheel-type robot in the duration of one hop determined in (a) above. Here too assume the maximum driving force that can be generated by the surface. c) How do the distances calculated in a) and b) above compare? Which assumption in your analysis would you say is most questionable? What would be your next step in this comparison?
Mech 220/04
11
FINAL/04
Mech 220/04
12
FINAL/04
Appendix
Mech 220/04
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