Problem Statement
A hydraulic press has a ram of 30 cm diameter and a plunger of 4.5 cm diameter. Find the weight lifted by the hydraulic press when the force applied at the plunger is 500 N.
Given Data
- Diameter of Ram, \(D = 30 \, \text{cm} = 0.3 \, \text{m}\)
- Diameter of Plunger, \(d = 4.5 \, \text{cm} = 0.045 \, \text{m}\)
- Force on Plunger, \(F = 500 \, \text{N}\)
Solution
1. Calculate the Area of the Ram (\(A\))
2. Calculate the Area of the Plunger (\(a\))
3. Calculate the Pressure Exerted by the Plunger
The pressure \(p\) created in the fluid by the force on the plunger is:
4. Calculate the Weight Lifted by the Ram (\(W\))
According to Pascal’s Law, this pressure is transmitted equally to the ram. The upward force on the ram (the weight it can lift) is this pressure multiplied by the ram’s area.
Converting to kilo-Newtons (kN):
The weight lifted by the hydraulic press is approximately \( W \approx 22,222 \, \text{N} \) or \( 22.22 \, \text{kN} \).
Explanation of Pascal’s Law
Pascal’s Law is the fundamental principle behind the hydraulic press. It states that a pressure change at any point in a confined, incompressible fluid is transmitted equally and undiminished to all points throughout the fluid.
In this system, the force \(F\) applied to the small plunger creates a pressure \(p = F/a\). Because the fluid is confined, this exact same pressure \(p\) is exerted on the bottom of the larger ram. Since the ram has a much larger area \(A\), the resulting upward force (\(W = p \times A\)) is significantly larger than the initial force \(F\).
Physical Meaning: Force Multiplication
The hydraulic press is a force-multiplying device. The calculation shows that a relatively small input force of 500 N (roughly the weight of a 51 kg person) can generate a massive lifting force of 22,222 N (roughly the weight of a 2265 kg car).
The mechanical advantage is the ratio of the output force to the input force, which is equal to the ratio of the areas:
$$ \text{Advantage} = \frac{W}{F} = \frac{A}{a} = \frac{0.07068}{0.00159} \approx 44.4 $$This means the press multiplies the applied force by more than 44 times. This principle is essential for heavy-duty applications like vehicle lifts, industrial forging presses, and aircraft hydraulic systems.
