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To determine the effort required to move 4000 lbs on the large piston in a hydraulic system, we can follow these steps:
1. Determine the Radii of the Pistons:
- For the small piston:
[tex]\[ \text{Diameter}_{\text{small}} = 1 \text{ inch} \][/tex]
[tex]\[ \text{Radius}_{\text{small}} = \frac{\text{Diameter}_{\text{small}}}{2} = \frac{1}{2} = 0.5 \text{ inches} \][/tex]
- For the large piston:
[tex]\[ \text{Diameter}_{\text{large}} = 10 \text{ inches} \][/tex]
[tex]\[ \text{Radius}_{\text{large}} = \frac{\text{Diameter}_{\text{large}}}{2} = \frac{10}{2} = 5 \text{ inches} \][/tex]
2. Calculate the Areas of the Pistons:
- The area [tex]\(A\)[/tex] of a circle is given by the formula:
[tex]\[ A = \pi \times (\text{radius})^2 \][/tex]
- For the small piston:
[tex]\[ \text{Area}_{\text{small}} = \pi \times (0.5)^2 \approx 0.7854 \text{ square inches} \][/tex]
- For the large piston:
[tex]\[ \text{Area}_{\text{large}} = \pi \times (5)^2 \approx 78.54 \text{ square inches} \][/tex]
3. Use the Principle of Hydraulics:
- According to Pascal's Principle, the force exerted on one piston is transmitted equally to the other piston through the hydraulic fluid.
- The effort required on the small piston to lift the load on the large piston can be found using the relationship:
[tex]\[ \frac{\text{Effort on small piston}}{\text{Area of small piston}} = \frac{\text{Force on large piston}}{\text{Area of large piston}} \][/tex]
- Rearranging the equation to solve for the effort on the small piston:
[tex]\[ \text{Effort}_{\text{small}} = \left(\frac{\text{Force}_{\text{large}} \times \text{Area}_{\text{small}}}{\text{Area}_{\text{large}}}\right) \][/tex]
- Plugging in the values:
[tex]\[ \text{Effort}_{\text{small}} = \left(\frac{4000 \text{ lbs} \times 0.7854 \text{ square inches}}{78.54 \text{ square inches}}\right) \][/tex]
[tex]\[ \text{Effort}_{\text{small}} \approx 40 \text{ lbs} \][/tex]
So, the effort that must be applied to the small piston to move 4000 lbs on the large piston is approximately 40 lbs.
1. Determine the Radii of the Pistons:
- For the small piston:
[tex]\[ \text{Diameter}_{\text{small}} = 1 \text{ inch} \][/tex]
[tex]\[ \text{Radius}_{\text{small}} = \frac{\text{Diameter}_{\text{small}}}{2} = \frac{1}{2} = 0.5 \text{ inches} \][/tex]
- For the large piston:
[tex]\[ \text{Diameter}_{\text{large}} = 10 \text{ inches} \][/tex]
[tex]\[ \text{Radius}_{\text{large}} = \frac{\text{Diameter}_{\text{large}}}{2} = \frac{10}{2} = 5 \text{ inches} \][/tex]
2. Calculate the Areas of the Pistons:
- The area [tex]\(A\)[/tex] of a circle is given by the formula:
[tex]\[ A = \pi \times (\text{radius})^2 \][/tex]
- For the small piston:
[tex]\[ \text{Area}_{\text{small}} = \pi \times (0.5)^2 \approx 0.7854 \text{ square inches} \][/tex]
- For the large piston:
[tex]\[ \text{Area}_{\text{large}} = \pi \times (5)^2 \approx 78.54 \text{ square inches} \][/tex]
3. Use the Principle of Hydraulics:
- According to Pascal's Principle, the force exerted on one piston is transmitted equally to the other piston through the hydraulic fluid.
- The effort required on the small piston to lift the load on the large piston can be found using the relationship:
[tex]\[ \frac{\text{Effort on small piston}}{\text{Area of small piston}} = \frac{\text{Force on large piston}}{\text{Area of large piston}} \][/tex]
- Rearranging the equation to solve for the effort on the small piston:
[tex]\[ \text{Effort}_{\text{small}} = \left(\frac{\text{Force}_{\text{large}} \times \text{Area}_{\text{small}}}{\text{Area}_{\text{large}}}\right) \][/tex]
- Plugging in the values:
[tex]\[ \text{Effort}_{\text{small}} = \left(\frac{4000 \text{ lbs} \times 0.7854 \text{ square inches}}{78.54 \text{ square inches}}\right) \][/tex]
[tex]\[ \text{Effort}_{\text{small}} \approx 40 \text{ lbs} \][/tex]
So, the effort that must be applied to the small piston to move 4000 lbs on the large piston is approximately 40 lbs.
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