mechanics 1 â mech 210 final examination
McGill University, Faculty of Engineering, Department Mechanical Engineering
MECHANICS 1 – MECH 210 FINAL EXAMINATION Tuesday April 17th 2005, 9:00-12:00 (3 hours)
Examiner: Prof. Francois Barthelat Associate Examiner: Prof. Pascal Hubert • • •
Closed book, Faculty approved calculator authorized The exam has 7 problems (100 points total) 1: An “enthusiastic” airplane mechanic (10 points) 2: The welded frame (10 points) 3: Three ways to open a bottle (15 points) 4: The hydraulic lift (13 points) 5: Beam or truss or… beam? (30 points) 6: Stabilize the rod (10 points) 7: The square plate (fluid statics) (12 points)
• • •
Each useful free body diagram is worth points. Indicate units in your results g = 9.81 m/s2
Page 1 of 7
Winter 2007
McGill University, Faculty of Engineering, Department Mechanical Engineering
Winter 2007
Problem 1: An “enthusiastic” airplane mechanics (10 points) (1) A 1800 kg airplane is on the ground. The centre of gravity is at point G (see figure below). The wheels at A and B are frictionless. What are the support reactions at A and B?
(2) A mechanics decides it’s time to check the engine. He locks the main wheels at B and starts the engine (the tail wheel A is not locked). The propeller generates a thrust T = 3000 N (see below). What are the support reaction at A and B now?
(3) The mechanics gets a little too “enthusiastic” and pushes the throttle further. Wheels at B are still locked, and after some point the plane tips and falls on its nose. At which thrust Tmax did this happen? (hint: the tail wheel A lost contact with the ground when the support reactions at A went to zero).
Page 2 of 7
McGill University, Faculty of Engineering, Department Mechanical Engineering
Winter 2007
Problem 2: The welded frame (10 points) (1) This welded frame is composed of two 100 lbs legs at 90°. It is attached to a wall by a ball and socket joint at A and by a bearing at B. A cable DC maintains the frame horizontal. What is the tension in the cable? The reactions at A and B do not need to be calculated.
Page 3 of 7
McGill University, Faculty of Engineering, Department Mechanical Engineering
Winter 2007
Problem 3: Three ways to open a bottle (15 points) The cork of this bottle requires a 300 N force for extraction. Determine the force(s) P needed to extract the cork using each of the following instruments:
1) The waiter’s corkscrew
2) The “acrobat” corkscrew: For this one you may want to write that the displacement at the periphery of the geared wheel is equal to the vertical displacement of the shaft.
3) The “accordion” corkscrew: does P depend on the length of the individual members?
Page 4 of 7
McGill University, Faculty of Engineering, Department Mechanical Engineering
Winter 2007
Problem 4: The hydraulic lift (13 points) A weight W is placed on the hydraulic lift showed below. A,D,E,H,G are pins, B and C are pins and rollers. Neglect the weight of the members. 1) Describe how you would calculate the force exerted by the hydraulic cylinder DH using conventional machine analysis techniques (draw FBDs, write number of equations, unknowns, explain how you would solve but do not solve) 2) Calculate the force exerted by cylinder DH using virtual work (hint: write the length DH as function of θ)
Page 5 of 7
McGill University, Faculty of Engineering, Department Mechanical Engineering
Winter 2007
Problem 5: Beam or truss or… beam? (30 points) A cantilever beam is used in a factory hall to support the forces showed below. The beam keeps failing at the point where it is attached on the wall. 1) Determine the bending and shear force diagrams of the beam. Why does is keep failing near the wall? 500 N
1000 N
Wall
0.5 m
1m
1m
2) Bob, an engineer in the factory, proposes we replace the beam with a truss structure. It is a little more expensive, but he claims this will fix the problem. Compute the internal forces in a few members (1 through 5) to check if Bob is right. 500 N
1000 N
Wall 0.5 m
0.5 m
0.5 m
0.5 m
0.5 m
0.5 m
3) Jim, another engineer, claims that a truss structure is too much effort: He says all we have to do is to attach the beam to the wall with a hinge and attach the free end to the ceiling with a cable (see diagram below). Determine the new shear and bending diagrams. Do you think this may fix the problem? cable 500 N
1000 N
Wall
0.5 m
1m
Page 6 of 7
1m
McGill University, Faculty of Engineering, Department Mechanical Engineering
Winter 2007
Problem 6: Stabilize the rod (10 points) (1) A rod of length L = 0.5m and weight m = 4 kg is attached to the floor by a frictionless pin (image 1). Assume that the weight is applied at the center of gravity of the rod. Compute θ for equilibrium, and determine if the equilibrium is stable. (2) In order to improve the stability, two identical springs are mounted at the top of the rod, as shown on image 2. The spring are at rest for θ =0. You can assume small angles so that the springs extension or compression is +/- Lθ. What is the minimum spring constant kmin to ensure stable equilibrium around θ =0? Image 1 Problem 7: Fluid statics (12 points) A square plate (size a = 0.5m) seals a square opening in a water tank (ρ =1000 kg/m3). The plate is hinged at point O. 1) Determine and draw the load distribution on the plate for h = 0 m, and h = 10m. 2) What is the total force exerted by the water on the plate for h = 0 m, and h = 10m? 3) What force P should be applied hold the plate in position for h = 0 m, and h = 10m? given formulae:
Page 7 of 7
Image 2
MECHANICS 1 – MECH 210 FINAL EXAMINATION Tuesday April 17th 2005, 9:00-12:00 (3 hours)
Examiner: Prof. Francois Barthelat Associate Examiner: Prof. Pascal Hubert • • •
Closed book, Faculty approved calculator authorized The exam has 7 problems (100 points total) 1: An “enthusiastic” airplane mechanic (10 points) 2: The welded frame (10 points) 3: Three ways to open a bottle (15 points) 4: The hydraulic lift (13 points) 5: Beam or truss or… beam? (30 points) 6: Stabilize the rod (10 points) 7: The square plate (fluid statics) (12 points)
• • •
Each useful free body diagram is worth points. Indicate units in your results g = 9.81 m/s2
Page 1 of 7
Winter 2007
McGill University, Faculty of Engineering, Department Mechanical Engineering
Winter 2007
Problem 1: An “enthusiastic” airplane mechanics (10 points) (1) A 1800 kg airplane is on the ground. The centre of gravity is at point G (see figure below). The wheels at A and B are frictionless. What are the support reactions at A and B?
(2) A mechanics decides it’s time to check the engine. He locks the main wheels at B and starts the engine (the tail wheel A is not locked). The propeller generates a thrust T = 3000 N (see below). What are the support reaction at A and B now?
(3) The mechanics gets a little too “enthusiastic” and pushes the throttle further. Wheels at B are still locked, and after some point the plane tips and falls on its nose. At which thrust Tmax did this happen? (hint: the tail wheel A lost contact with the ground when the support reactions at A went to zero).
Page 2 of 7
McGill University, Faculty of Engineering, Department Mechanical Engineering
Winter 2007
Problem 2: The welded frame (10 points) (1) This welded frame is composed of two 100 lbs legs at 90°. It is attached to a wall by a ball and socket joint at A and by a bearing at B. A cable DC maintains the frame horizontal. What is the tension in the cable? The reactions at A and B do not need to be calculated.
Page 3 of 7
McGill University, Faculty of Engineering, Department Mechanical Engineering
Winter 2007
Problem 3: Three ways to open a bottle (15 points) The cork of this bottle requires a 300 N force for extraction. Determine the force(s) P needed to extract the cork using each of the following instruments:
1) The waiter’s corkscrew
2) The “acrobat” corkscrew: For this one you may want to write that the displacement at the periphery of the geared wheel is equal to the vertical displacement of the shaft.
3) The “accordion” corkscrew: does P depend on the length of the individual members?
Page 4 of 7
McGill University, Faculty of Engineering, Department Mechanical Engineering
Winter 2007
Problem 4: The hydraulic lift (13 points) A weight W is placed on the hydraulic lift showed below. A,D,E,H,G are pins, B and C are pins and rollers. Neglect the weight of the members. 1) Describe how you would calculate the force exerted by the hydraulic cylinder DH using conventional machine analysis techniques (draw FBDs, write number of equations, unknowns, explain how you would solve but do not solve) 2) Calculate the force exerted by cylinder DH using virtual work (hint: write the length DH as function of θ)
Page 5 of 7
McGill University, Faculty of Engineering, Department Mechanical Engineering
Winter 2007
Problem 5: Beam or truss or… beam? (30 points) A cantilever beam is used in a factory hall to support the forces showed below. The beam keeps failing at the point where it is attached on the wall. 1) Determine the bending and shear force diagrams of the beam. Why does is keep failing near the wall? 500 N
1000 N
Wall
0.5 m
1m
1m
2) Bob, an engineer in the factory, proposes we replace the beam with a truss structure. It is a little more expensive, but he claims this will fix the problem. Compute the internal forces in a few members (1 through 5) to check if Bob is right. 500 N
1000 N
Wall 0.5 m
0.5 m
0.5 m
0.5 m
0.5 m
0.5 m
3) Jim, another engineer, claims that a truss structure is too much effort: He says all we have to do is to attach the beam to the wall with a hinge and attach the free end to the ceiling with a cable (see diagram below). Determine the new shear and bending diagrams. Do you think this may fix the problem? cable 500 N
1000 N
Wall
0.5 m
1m
Page 6 of 7
1m
McGill University, Faculty of Engineering, Department Mechanical Engineering
Winter 2007
Problem 6: Stabilize the rod (10 points) (1) A rod of length L = 0.5m and weight m = 4 kg is attached to the floor by a frictionless pin (image 1). Assume that the weight is applied at the center of gravity of the rod. Compute θ for equilibrium, and determine if the equilibrium is stable. (2) In order to improve the stability, two identical springs are mounted at the top of the rod, as shown on image 2. The spring are at rest for θ =0. You can assume small angles so that the springs extension or compression is +/- Lθ. What is the minimum spring constant kmin to ensure stable equilibrium around θ =0? Image 1 Problem 7: Fluid statics (12 points) A square plate (size a = 0.5m) seals a square opening in a water tank (ρ =1000 kg/m3). The plate is hinged at point O. 1) Determine and draw the load distribution on the plate for h = 0 m, and h = 10m. 2) What is the total force exerted by the water on the plate for h = 0 m, and h = 10m? 3) What force P should be applied hold the plate in position for h = 0 m, and h = 10m? given formulae:
Page 7 of 7
Image 2
