Problem Statement
A hydraulic press has a ram of 30 cm diameter and a plunger of 5 cm diameter. Find the weight lifted by the hydraulic press when the force applied at the plunger is 400 N.
Given Data
- Diameter of ram, \(D = 30 \, \text{cm} = 0.3 \, \text{m}\)
- Diameter of plunger, \(d = 5 \, \text{cm} = 0.05 \, \text{m}\)
- Force applied at plunger, \(F = 400 \, \text{N}\)
Solution
1. Calculate Piston Areas
First, we calculate the area of the plunger (\(a\)) and the ram (\(A\)).
2. Apply Pascal’s Law
According to Pascal’s Law, the pressure exerted on the plunger is transmitted equally to the ram. Let \(W\) be the weight lifted by the ram.
3. Solve for the Weight Lifted (W)
Rearrange the formula to solve for \(W\) and substitute the known values.
The weight can also be expressed in kilonewtons (kN).
The weight lifted by the hydraulic press is \( 14400 \, \text{N} \) or \( 14.4 \, \text{kN} \).
Explanation of Pascal’s Law
Pascal’s Law is a fundamental principle in fluid mechanics that states a pressure change at any point in a confined incompressible fluid is transmitted throughout the fluid such that the same change occurs everywhere. In a hydraulic system, this means the pressure created by the small force on the plunger (\(P = F/a\)) is the same pressure that acts on the large ram.
Because this pressure (\(P\)) acts on a much larger area (\(A\)) at the ram, the resulting upward force (\(W = P \times A\)) is much larger than the initial force (\(F\)). This phenomenon is known as force multiplication.
Physical Meaning
The result demonstrates the powerful advantage of a hydraulic press. A relatively small input force of 400 N (roughly the weight of a 40 kg object) applied to the 5 cm plunger is magnified to lift a massive weight of 14,400 N (roughly the weight of a 1468 kg object, like a car).
The force multiplication factor is equal to the ratio of the areas (\(A/a\)), which in this case is 36. This means the output force is 36 times greater than the input force, showcasing how hydraulic systems provide a significant mechanical advantage.
