You are using a hydraulic jack to lift a 12000 Newton object. This input piston of your jack has a radius of 0.85 meters and you can exert a maximum of 3500 Newtons on it. How large does the radius of the output piston need to be in order to lift this object without failing?
R^2 * Force = constant
(same number for input and output pistons)
R^2 * 12000 = (0.85)^2 * 3500
Solve for output piston radius, R
To determine the radius of the output piston needed to lift the object without failing, we can use the principle of Pascal's law, which states that the pressure in a fluid is transmitted equally in all directions.
First, let's calculate the pressure applied by the input piston:
Pressure = Force / Area
The area of the input piston can be calculated using the formula:
Area = π * radius^2
Given that the radius of the input piston is 0.85 meters and the maximum force that can be exerted is 3500 Newtons, the pressure applied by the input piston can be calculated as follows:
Area = π * (0.85)^2 ≈ 2.27 square meters
Pressure = 3500 Newtons / 2.27 square meters ≈ 1540 Pascal
Now, since the pressure is transmitted equally, we can equate the pressure applied by the input piston to the pressure exerted by the output piston:
Pressure = Force / Area
Given that the force exerted by the output piston is 12,000 Newtons, and we need to find the radius of the output piston, let's rearrange the formula:
Area = Force / Pressure
Substituting the values, we have:
Area = 12,000 Newtons / 1540 Pascal ≈ 7.79 square meters
Finally, we can calculate the radius of the output piston using the formula:
Radius = √(Area / π)
Radius = √(7.79 / π) ≈ 1.40 meters
Therefore, the radius of the output piston needs to be approximately 1.40 meters in order to lift the 12,000 Newton object without failing.