Final answer:
To use the weight of an 80-kg person to lift a 2100-kg elephant with a hydraulic lift, the minimum diameter of the piston under the elephant must be approximately 1.54 meters. This calculation is based on the conservation of pressure and the relationship between force and area in a hydraulic system.
Step-by-step explanation:
To solve this problem, we use the principle of a hydraulic lift, which operates based on Pascal's law stating that a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container. The hydraulic lift system allows a person to lift a heavy object such as an elephant by applying a force on a smaller piston, which then exerts a greater force on the larger piston where the elephant stands, due to the difference in areas.
First, we calculate the force the person exerts due to their weight. This is equal to their mass times the acceleration due to gravity (F = m * g). For the person with a mass of 80 kg, assuming g = 9.81 m/s2, this force is:
FA = 80 kg * 9.81 m/s2
= 784.8 N.
Next, since the pressure (P) is the same on both sides and is equal to the force divided by the area (P = F/A), we find the area of piston A, which is a circle with diameter 0.30 m and thus a radius of 0.15 m. Using the formula for the area of a circle (A = π * r2), we get:
AA = π * (0.15 m)2
= 0.0707 m2.
The pressure applied by the person on piston A is:
P = 784.8 N / 0.0707 m2
= 11101.8 Pa.
The same pressure must be applied by the elephant on piston B to lift it. The weight of the elephant is its mass times the acceleration due to gravity:
FB = 2100 kg * 9.81 m/s2
= 20601 N.
To find the minimum area of piston B required to provide this force with the existing pressure, we rearrange the pressure formula:
AB = FB / P
= 20601 N / 11101.8 Pa
= 1.855 m2.
Finally, we calculate the diameter of piston B, which is twice the radius:
d = 2 * √(AB / π)
= 2 * √(1.855 m2 / π)
= 2 * √(0.590 m2)
≈ 1.54 m.
The minimum diameter of piston B should be approximately 1.54 meters to lift the elephant using the weight of the person.