1) Design a gear system using 12, 34 and 36 teeth gears that achieve a gear ratio of 13.5:1 and has the top and bottom gear spinning in opposite directions.
2) For a pulley system with 8 supporting strands, how far would you have to pull the rope to lift a 125 kg object 1.0 meters off the ground? What if the pulley system were only 80% efficient?
To design a gear system with a gear ratio of 13.5:1 and the top and bottom gears spinning in opposite directions using gears with 12, 34, and 36 teeth, we can follow these steps:
1) Calculate the gear ratios of the gear pairs: In a gear system, the gear ratio is determined by the ratio of the number of teeth on the driving gear to the number of teeth on the driven gear. Calculate the gear ratios for the gear pairings as follows:
- Top gear (12 teeth) to driving gear (34 teeth): 34/12 = 2.83
- Driving gear (34 teeth) to bottom gear (36 teeth): 36/34 = 1.06
2) Determine the combination of gears that achieves the desired gear ratio: Since we want a gear ratio of 13.5:1, we need to find a combination of gear pairings that multiply to approximately 13.5.
To achieve this, we can multiply the gear ratios of the gear pairs together:
2.83 * 1.06 = 2.998
This gives us a gear ratio of approximately 3:1.
3) Adjust the gear ratio: Since we need a gear ratio of 13.5:1, we need to increase the gear ratio.
To achieve this, we can add additional gears in series by using an idler gear. An idler gear is a gear that is placed between two gears but does not affect the overall gear ratio.
To increase the gear ratio from 3:1 to 13.5:1, we can introduce an idler gear with the desired number of teeth. Let's assume the idler gear has 45 teeth.
The modified gear system will be as follows:
- Top gear (12 teeth) to driving gear (34 teeth)
- Driving gear (34 teeth) to idler gear (45 teeth)
- Idler gear (45 teeth) to bottom gear (36 teeth)
The resulting gear ratio is (34/12) * (45/34) * (36/45) = 13.5:1.
Please note that using specific gear sizes might require additional considerations, such as the gear meshing properly and sufficient space for the gears to fit together.
Regarding the second question:
To calculate the distance you need to pull the rope in a pulley system with 8 supporting strands, we need some additional information:
- Efficiency of the pulley system: Efficiency is the ratio of the output work to the input work. In this case, it represents how much of the input force is converted into useful work. Let's assume the pulley system is 80% efficient.
1) Calculate the mechanical advantage of the pulley system: The mechanical advantage (MA) of a pulley system with multiple supporting strands is equal to the number of supporting strands. In this case, the MA is 8.
2) Determine the force required to lift the object: To lift the 125 kg object, we need to calculate the force required. The formula for force (F) is: F = mass * gravity, where gravity is approximately 9.8 m/s^2.
F = 125 kg * 9.8 m/s^2 = 1225 N
3) Calculate the work done to lift the object: Work (W) is the product of force and distance. In this case, the distance is 1.0 meters.
W = F * d = 1225 N * 1.0 m = 1225 Joules
4) Adjust for efficiency: If the pulley system is 80% efficient, we need to account for the loss of energy. Multiply the work done by 1/efficiency (1/0.8) to determine the actual work required.
Actual Work = W / efficiency = 1225 Joules / 0.8 = 1531.25 Joules
Therefore, you would need to pull the rope a distance that corresponds to 1531.25 Joules of work to lift the 125 kg object 1.0 meter off the ground using a pulley system with 8 supporting strands and 80% efficiency.