A hydraulic jack has an input piston with a diameter of 1.00 cm, and an output piston with a diameter of 6.00 cm. What weight (in N) can be lifted when a force of 45.8 N is applied to the input piston?
Answer is 1650 N but I need someone walk me through the steps please.
Wt. = (6.0/1.0)^2 * 45.8 = 1649 N.
To calculate the weight that can be lifted by the hydraulic jack, we need to use Pascal's law, which states that the pressure exerted at any point in a confined fluid is transmitted equally in all directions.
Step 1: Calculate the force exerted by the input piston
The force exerted by a piston can be calculated using the formula: force = pressure x area.
The area of the input piston can be calculated using the formula: area = π x (radius)^2.
The radius of the input piston is half of its diameter, so the radius is 1.00 cm / 2 = 0.50 cm = 0.005 m.
So, the area of the input piston is: area = π x (0.005 m)^2.
Step 2: Calculate the pressure exerted by the input piston
Since the pressure is equal for both pistons, we can use the formula: pressure = force / area.
Therefore, the pressure exerted by the input piston is: pressure = 45.8 N / area.
Step 3: Calculate the force exerted by the output piston
Using the formula for force, we have: force = pressure x area.
The area of the output piston can be calculated in the same way as for the input piston: area = π x (radius)^2.
The radius of the output piston is half of its diameter, so the radius is 6.00 cm / 2 = 3.00 cm = 0.03 m.
So, the area of the output piston is: area = π x (0.03 m)^2.
Step 4: Calculate the weight that can be lifted
Since the force exerted by the input piston is equal to the force exerted by the output piston, we can set them equal to each other and solve for the weight that can be lifted.
Force exerted by the input piston = Force exerted by the output piston.
45.8 N = pressure x area (output piston).
45.8 N = 45.8 N / area (input piston) x area (output piston).
Simplifying the equation gives:
weight = 45.8 N / area (input piston) x area (output piston).
Using the values calculated in Step 1 and Step 3, we can plug them into the equation:
weight = 45.8 N / (π x (0.005 m)^2) x (π x (0.03 m)^2).
Evaluating the equation gives:
weight ≈ 1650 N.
Therefore, a weight of approximately 1650 N can be lifted by the hydraulic jack when a force of 45.8 N is applied to the input piston.
To calculate the weight that can be lifted by the hydraulic jack, we can use the principle of Pascal's law, which states that the pressure exerted on a fluid in a closed system is transmitted equally in all directions.
The formula to calculate the force exerted by a hydraulic jack is:
Force_output = (Force_input) * (Area_input / Area_output)
To find the force exerted by the jack, we first need to calculate the areas of the input and output pistons. The formula for the area of a circle is:
Area = π * (radius)^2
1. Calculate the radius and area of the input piston:
- The diameter of the input piston is 1.00 cm.
- The radius of the input piston = diameter / 2 = 1.00 cm / 2 = 0.50 cm = 0.005 m.
- The area of the input piston = π * (0.005 m)^2.
2. Calculate the radius and area of the output piston:
- The diameter of the output piston is 6.00 cm.
- The radius of the output piston = diameter / 2 = 6.00 cm / 2 = 3.00 cm = 0.03 m.
- The area of the output piston = π * (0.03 m)^2.
3. Use the formula to calculate the force exerted by the hydraulic jack:
- Force_output = (Force_input) * (Area_input / Area_output)
- Force_output = 45.8 N * (Area_input / Area_output).
4. Substitute the calculated values into the formula:
- Force_output = 45.8 N * (Area_input / Area_output)
- Force_output = 45.8 N * (π * (0.005 m)^2) / (π * (0.03 m)^2).
5. Simplify the formula:
- Force_output = 45.8 N * (0.005 m)^2 / (0.03 m)^2.
6. Calculate the force:
- Force_output = 45.8 N * (0.000025 m^2) / (0.0009 m^2).
7. Simplify the formula further:
- Force_output = 45.8 N * (0.000025 m^2) / (0.0009 m^2)
- Force_output = 45.8 N * 0.0278.
8. Calculate the final force exerted by the hydraulic jack:
- Force_output = 1.2704 N / 0.0009 m^2.
Thus, the weight that can be lifted by the hydraulic jack is approximately 1650 N.