A hydraulic device is used to lift the bucket of a backhoe. The input is a round cylinder that is 1.0 cm in radius. The output piston is round and 6.0 cm in radius. How much can it lift if the input force is 500 newtons?
so converting radius to Area A=Pi x r^2
so I have Fi=500N Ai= 500cm^2
Fo=? Ao=18000cm^2
Fi/Ai=Fo/Ao
Answer: Fo= 18000N
Am I correct?
looks good
Well, let's take a closer look at the problem. Your conversion of radius to area is correct, but your calculation of the areas seems to be off. The area of the input cylinder would be π(1.0 cm)^2, which is approximately 3.14 square centimeters. The area of the output piston would be π(6.0 cm)^2, which is approximately 113.04 square centimeters.
Now, let's use the formula Fi/Ai = Fo/Ao to solve for Fo, the output force. Plugging in the values we have:
500 N / 3.14 cm² = Fo / 113.04 cm²
Simplifying this equation, we find:
Fo ≈ (500 N * 113.04 cm²) / 3.14 cm² ≈ 17,992.04 N
So, it seems like you made a calculation error. The correct answer would be approximately 17,992.04 newtons, not 18,000 newtons. Keep in mind that this is an idealized scenario and does not consider factors such as friction and other real-world limitations.
Yes, your calculations are correct. To find the output force, you can use the formula Fi/Ai = Fo/Ao, where Fi is the input force, Ai is the input area, Fo is the output force, and Ao is the output area.
Given that the radius of the input cylinder is 1.0 cm, the input area can be calculated as follows:
Ai = π × (1.0 cm)^2 = 3.14 cm^2
Similarly, with the radius of the output piston being 6.0 cm, the output area can be calculated as:
Ao = π × (6.0 cm)^2 = 113.04 cm^2
Substituting these values into the equation:
500 N / 3.14 cm^2 = Fo / 113.04 cm^2
Solving for Fo:
Fo = 500 N × (113.04 cm^2 / 3.14 cm^2) ≈ 18000 N
So, the hydraulic device can lift approximately 18000 newtons if the input force is 500 newtons.
Yes, you are correct!
To solve this problem, you used the principle of Pascal's law, which states that the pressure applied to an enclosed fluid is transmitted uniformly in all directions. This is the basis for hydraulic systems.
First, you calculated the cross-sectional area of the input piston. Given that the radius is 1.0 cm, you correctly used the formula A = πr^2 to find the input piston area: Ai = π(1.0 cm)^2 = 3.14 cm^2.
Next, you calculated the force exerted on the input piston. You were given that the force is 500 N, so Fi = 500 N.
Similarly, you found the cross-sectional area of the output piston. The radius is 6.0 cm, so Ao = π(6.0 cm)^2 = 113.04 cm^2.
Now, you used the principle of Pascal's law, which states that the ratio of the force over the area remains constant in a hydraulic system. Therefore, Fi/Ai = Fo/Ao.
Plugging in the known values, you have 500 N / 3.14 cm^2 = Fo / 113.04 cm^2.
Simplifying this equation, you find: Fo = (500 N / 3.14 cm^2) × 113.04 cm^2 = 18011.42 N.
Rounded to the nearest whole number, the output force is 18000 N.
Great job on your calculations!