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Understanding Gear Motion and Spider-Fly Distance Problem in Engineering

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tamika
15 diapositives · 330 Views · Sep 08, 2014

This document explores two engineering problems involving mechanics and spatial reasoning. The first part analyzes the rotation of two gears, detailing the final orientation of a gear tooth and the total rotations during the motion. The second part presents a scenario involving a spider and a fly in a room, calculating the minimum distance the spider must travel to reach the fly, both when starting on the ceiling and when on the floor. Each problem is approached with mathematical principles and problem-solving strategies, encouraging visualization of spatial relationships.

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  1. September 9, 2011 IOTPOLYENGINEERING I1-7 DRILL A: Two gears, A and B, are placed side by side. Gear A is rolled over gear B as shown. When gear A reaches the opposite side of gear B, which way will the orange gear tooth face? If gear A continues to roll around gear B until it returns to the original position, how many times will gear A have rotated around its own center? A B

  2. DRILL A: Two Gears – SIMULATION SOLUTION IOTPOLYENGINEERING I1-7 When gear A reaches the opposite side of gear B, the orange gear tooth will be facing up, the same way it started. If gear A continues to roll around gear B until it returns to the original position, it will have rotated around its own center twice.

  3. IOTPOLYENGINEERING I1-7 PROBLEM #13 - HOMEWORK : SPIDER & FLY A Given: A spider and a fly are in a room whose dimensions are 25 feet wide by 15 feet deep by 8 feet high. The spider is on the CEILING and the fly is on the FLOOR. If one corner of the room represents the origin (0,0,0) of an x-y-z coordinate system, then the spider is located at (20,8,-11 ) and the fly is located at (5,0,-7 ). See the given diagram. Problem: What is the MINIMUM DISTANCE that the spider must travel to reach the fly on the floor?

  4. IOTPOLYENGINEERING I1-7 PROBLEM #13 - HOMEWORK : SPIDER & FLY A - SOLUTION The spider is on the CEILING and the fly is on the FLOOR, but the spider is not directly above the fly. The shortest distance between the spider and the fly is a STRAIGHT LINE of about 17.5 feet, but the spider cannot travel that straight line because the spider cannot fly directly toward the fly on the floor. However, the spider can drop straight down to the floor by its spider silk. The distance to the floor is 8 feet. After the spider reaches the floor, its coordinates will be (20,0,-11). The following diagram shows both the spider and fly on the floor.

  5. IOTPOLYENGINEERING I1-7 The Pythagorean Theorem can be used to find the distance between the spider and fly . SUMMARY: The spider would need to travel 8 feet DOWN from the ceiling and 15.5 feet ACROSS the floor for a TOTAL distance of 23.5 feet.

  6. IOTPOLYENGINEERING I1-7 PROBLEM #13 - HOMEWORK : SPIDER & FLY B Given: A spider and a fly are in a room whose dimensions are 25 feet wide by 15 feet deep by 8 feet high. The spider is on the FLOOR and the fly is on the CEILING. If one corner of the room represents the origin (0,0,0) of an x-y-z coordinate system, then the spider is located at (5,0,-7) and the fly is located at (20,8,-11 ). See the given diagram. Problem: What is the MINIMUM DISTANCE that the spider must travel to reach the fly on the ceiling?

  7. IOTPOLYENGINEERING I1-7 PROBLEM #13 - HOMEWORK : SPIDER & FLY B - SOLUTION The spider is on the FLOOR and the fly is on the CEILING. The shortest distance between the spider and the fly is a STRAIGHT LINE of about 17.5 feet, but the spider cannot travel that straight line because the spider cannot fly directly toward the fly on the ceiling. The spider will have to travel across the floor, go up a wall, and then travel across the ceiling to the fly. 1. What path should the spider travel toward a wall? 2. Which wall should the spider climb? 3. What path should the spider travel up the wall? 4. What path should the spider travel across the ceiling?

  8. IOTPOLYENGINEERING I1-7 PROBLEM #13 - HOMEWORK : SPIDER & FLY B - SOLUTION 1. What path should the spider travel toward a wall? 2. Which wall should the spider climb? 3. What path should the spider travel up the wall? 4. What path should the spider travel across the ceiling? • Traveling straight and perpendicular toward the closest wall seems logical, but it won’t give the shortest total path. • The walls are all 8 feet high so it seems that no wall is better than another. Actually, one wall is better than the rest. • Traveling straight up the wall seems logical, but it won’t give the shortest total path. • 4. Once the spider reaches the ceiling, it should travel directly toward the fly. This seems logical, and is correct.

  9. This problem can be solved by unfolding the room. IOTPOLYENGINEERING I1-7

  10. At home, use a scissors to cut your paper as follows: IOTPOLYENGINEERING I1-7 TAPE Determine the location of the Fly on all 4 ceilings views.

  11. IOTPOLYENGINEERING I1-7 This model will help visualize the problem.

  12. When the room is folded flat, we can see that the strategy of “directly to a wall, up the wall, and across the ceiling” does NOT give the shortest possible path! They aren’t straight. IOTPOLYENGINEERING I1-7

  13. Drawing straight lines from the spider’s location to the four fly locations gives shorter paths in all 4 cases, and one of these is the shortest possible path! IOTPOLYENGINEERING I1-7

  14. IOTPOLYENGINEERING I1-7 2.5 inches represents 25 feet.

  15. September 9, 2011 IOTPOLYENGINEERING I1-7 HOMEWORK: Spider & Fly Problem Letter your answers. Use complete sentences. 1. What mathematical principles were involved? 2. What problem solving strategies were used? 3. What did you learn by solving this problem? 4. Describe in detail the path the spider must take from the floor to the ceiling.