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Sagot :
Let's go through the table and calculate the required ratios and output RPM for each gear system:
### Table Details:
1. Input (effort): The force or torque applied to the input gear.
2. Output (load): The force or torque exerted by the output gear.
3. Input RPM: The rotational speed of the input gear.
4. Ratio: The ratio of the output load to the input effort.
5. Output RPM: The resulting rotational speed of the output gear.
### Step-by-Step Calculations:
#### For Gear System 1:
- Input (effort): 25
- Output (load): 75
- Input RPM: 120
Ratio:
[tex]\[ \text{Ratio} = \frac{\text{Output (load)}}{\text{Input (effort)}} = \frac{75}{25} = 3.0 \][/tex]
Output RPM:
[tex]\[ \text{Output RPM} = \frac{\text{Input RPM}}{\text{Ratio}} = \frac{120}{3.0} = 40.0 \][/tex]
#### For Gear System 2:
- Input (effort): 20
- Output (load): 120
- Input RPM: 120
Ratio:
[tex]\[ \text{Ratio} = \frac{\text{Output (load)}}{\text{Input (effort)}} = \frac{120}{20} = 6.0 \][/tex]
Output RPM:
[tex]\[ \text{Output RPM} = \frac{\text{Input RPM}}{\text{Ratio}} = \frac{120}{6.0} = 20.0 \][/tex]
#### For Gear System 3:
- Input (effort): 80
- Output (load): 20
- Input RPM: 200
Ratio:
[tex]\[ \text{Ratio} = \frac{\text{Output (load)}}{\text{Input (effort)}} = \frac{20}{80} = 0.25 \][/tex]
Output RPM:
[tex]\[ \text{Output RPM} = \frac{\text{Input RPM}}{\text{Ratio}} = \frac{200}{0.25} = 800.0 \][/tex]
#### For Gear System 4:
- Input (effort): 40
- Output (load): 200
- Input RPM: 150
Ratio:
[tex]\[ \text{Ratio} = \frac{\text{Output (load)}}{\text{Input (effort)}} = \frac{200}{40} = 5.0 \][/tex]
Output RPM:
[tex]\[ \text{Output RPM} = \frac{\text{Input RPM}}{\text{Ratio}} = \frac{150}{5.0} = 30.0 \][/tex]
#### For Gear System 5:
- Input (effort): 75
- Output (load): 25
- Input RPM: 180
Ratio:
[tex]\[ \text{Ratio} = \frac{\text{Output (load)}}{\text{Input (effort)}} = \frac{25}{75} = 0.333 \][/tex]
Output RPM:
[tex]\[ \text{Output RPM} = \frac{\text{Input RPM}}{\text{Ratio}} = \frac{180}{0.333} \approx 540.0 \][/tex]
### Completed Table:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline \text{Input (effort)} & \text{Output (load)} & \text{Input RPM} & \text{Ratio} & \text{Output RPM} \\ \hline 25 & 75 & 120 & 3.0 & 40.0 \\ \hline 20 & 120 & 120 & 6.0 & 20.0 \\ \hline 80 & 20 & 200 & 0.25 & 800.0 \\ \hline 40 & 200 & 150 & 5.0 & 30.0 \\ \hline 75 & 25 & 180 & 0.333 & 540.0 \\ \hline \end{array} \][/tex]
This table includes the calculated ratios and output RPMs for each gear system.
### Table Details:
1. Input (effort): The force or torque applied to the input gear.
2. Output (load): The force or torque exerted by the output gear.
3. Input RPM: The rotational speed of the input gear.
4. Ratio: The ratio of the output load to the input effort.
5. Output RPM: The resulting rotational speed of the output gear.
### Step-by-Step Calculations:
#### For Gear System 1:
- Input (effort): 25
- Output (load): 75
- Input RPM: 120
Ratio:
[tex]\[ \text{Ratio} = \frac{\text{Output (load)}}{\text{Input (effort)}} = \frac{75}{25} = 3.0 \][/tex]
Output RPM:
[tex]\[ \text{Output RPM} = \frac{\text{Input RPM}}{\text{Ratio}} = \frac{120}{3.0} = 40.0 \][/tex]
#### For Gear System 2:
- Input (effort): 20
- Output (load): 120
- Input RPM: 120
Ratio:
[tex]\[ \text{Ratio} = \frac{\text{Output (load)}}{\text{Input (effort)}} = \frac{120}{20} = 6.0 \][/tex]
Output RPM:
[tex]\[ \text{Output RPM} = \frac{\text{Input RPM}}{\text{Ratio}} = \frac{120}{6.0} = 20.0 \][/tex]
#### For Gear System 3:
- Input (effort): 80
- Output (load): 20
- Input RPM: 200
Ratio:
[tex]\[ \text{Ratio} = \frac{\text{Output (load)}}{\text{Input (effort)}} = \frac{20}{80} = 0.25 \][/tex]
Output RPM:
[tex]\[ \text{Output RPM} = \frac{\text{Input RPM}}{\text{Ratio}} = \frac{200}{0.25} = 800.0 \][/tex]
#### For Gear System 4:
- Input (effort): 40
- Output (load): 200
- Input RPM: 150
Ratio:
[tex]\[ \text{Ratio} = \frac{\text{Output (load)}}{\text{Input (effort)}} = \frac{200}{40} = 5.0 \][/tex]
Output RPM:
[tex]\[ \text{Output RPM} = \frac{\text{Input RPM}}{\text{Ratio}} = \frac{150}{5.0} = 30.0 \][/tex]
#### For Gear System 5:
- Input (effort): 75
- Output (load): 25
- Input RPM: 180
Ratio:
[tex]\[ \text{Ratio} = \frac{\text{Output (load)}}{\text{Input (effort)}} = \frac{25}{75} = 0.333 \][/tex]
Output RPM:
[tex]\[ \text{Output RPM} = \frac{\text{Input RPM}}{\text{Ratio}} = \frac{180}{0.333} \approx 540.0 \][/tex]
### Completed Table:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline \text{Input (effort)} & \text{Output (load)} & \text{Input RPM} & \text{Ratio} & \text{Output RPM} \\ \hline 25 & 75 & 120 & 3.0 & 40.0 \\ \hline 20 & 120 & 120 & 6.0 & 20.0 \\ \hline 80 & 20 & 200 & 0.25 & 800.0 \\ \hline 40 & 200 & 150 & 5.0 & 30.0 \\ \hline 75 & 25 & 180 & 0.333 & 540.0 \\ \hline \end{array} \][/tex]
This table includes the calculated ratios and output RPMs for each gear system.
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