To solve this problem, we need to apply the principles of hydraulics and leverage. A hydraulic press operates based on Pascal's principle, which states that pressure applied to a confined fluid is transmitted undiminished throughout the fluid. Additionally, we will use the concept of levers to determine the total force produced on the larger piston. Let’s break it down step by step.
Understanding the Components
We have two pistons with diameters of 0.04 m (the smaller piston) and 0.40 m (the larger piston). The ratio of the lever arms is 6:1, meaning that for every 6 units of distance the force is applied on the longer arm, the smaller piston moves 1 unit of distance.
Calculating Areas of the Pistons
The area of a piston is crucial because the force exerted by the hydraulic fluid depends on the area. The area \(A\) of a circle is calculated using the formula:
A = πr²
Where \(r\) is the radius of the piston. First, we need to find the radius of both pistons:
- Radius of the smaller piston: \(r_1 = \frac{0.04}{2} = 0.02 \, \text{m}\)
- Radius of the larger piston: \(r_2 = \frac{0.40}{2} = 0.20 \, \text{m}\)
Now, we can calculate the areas:
- Area of the smaller piston:
A_1 = π(0.02)² ≈ 0.00125664 \, \text{m}²
- Area of the larger piston:
A_2 = π(0.20)² ≈ 0.1256637 \, \text{m}²
Applying the Lever Principle
Next, we need to consider the lever. The force applied at the end of the longer arm is 75 kgf. To convert this force into Newtons, we use the conversion factor \(1 \, \text{kgf} = 9.81 \, \text{N}\):
Force applied (F) = 75 \, \text{kgf} × 9.81 \, \text{N/kgf} ≈ 735.75 \, \text{N}
Since the lever arms are in a 6:1 ratio, the force exerted on the smaller piston can be calculated using the ratio of the lever arms:
Force on smaller piston (F_1) = F × 6 = 735.75 \, \text{N} × 6 ≈ 4414.5 \, \text{N}
Calculating the Force on the Larger Piston
Now, using the principle of hydraulics, we know that the pressure exerted on both pistons is the same:
Pressure (P) = Force (F) / Area (A)
For the smaller piston:
P = F_1 / A_1 = 4414.5 \, \text{N} / 0.00125664 \, \text{m}² ≈ 3511000.2 \, \text{Pa}
Now, we can find the force exerted on the larger piston using the same pressure:
F_2 = P × A_2 = 3511000.2 \, \text{Pa} × 0.1256637 \, \text{m}² ≈ 4414.5 \, \text{N}
Final Calculation
Thus, the total force produced on the larger piston is approximately:
F_2 ≈ 4414.5 \, \text{N} ≈ 450 \, \text{kgf}
In summary, when a force of 75 kgf is applied at the end of the longer arm of the lever, the total force produced on the larger piston is about 450 kgf. This illustrates the power of hydraulic systems and levers in amplifying force effectively.
