To solve the problems, we will need to calculate the volumes and forces using the relationships in hydraulic systems.
2.1.1 Calculate the Volume of Liquid per Stroke
The volume of liquid displaced per stroke in a hydraulic press can be calculated using the volume formula for a cylinder:
\[ V = A \cdot h \]
where:
- \( V \) is the volume
- \( A \) is the cross-sectional area of the cylinder
- \( h \) is the height (or stroke length)
First, we need to calculate the cross-sectional area of the plunger and then use the stroke length.
Step 1: Calculate the area of the plunger
The area \( A \) can be calculated as: \[ A = \pi \left( \frac{d}{2} \right)^2 \]
where \( d \) is the diameter.
For the plunger: \( d_{plunger} = 55 , \text{mm} = 0.055 , \text{m} \)
\[ A_{plunger} = \pi \left( \frac{0.055}{2} \right)^2 = \pi \left( 0.0275 \right)^2 \approx 0.002375 , \text{m}^2 \]
Step 2: Calculate the volume per stroke
Height (stroke length) \( h = 106 , \text{mm} = 0.106 , \text{m} \)
\[ V = A_{plunger} \cdot h = 0.002375 , \text{m}^2 \cdot 0.106 , \text{m} \approx 0.00025125 , \text{m}^3 \]
So, the volume of liquid per stroke is approximately: \[ \text{Volume} \approx 0.000251 , \text{m}^3 , (or , 251 , cm^3) \]
2.1.2 Calculate the Force Exerted by the Ram
The force exerted by the ram can be calculated using Pascal's principle, which states that the pressure applied to a confined fluid is transmitted undiminished in all directions.
Step 1: Calculate the force exerted on the plunger
Given the force applied on the plunger \( F_{plunger} = 75 , \text{N} \).
Step 2: Calculate the pressure in the system
Pressure \( P \) generated by the plunger is given by: \[ P = \frac{F_{plunger}}{A_{plunger}} \]
Calculate the pressure: \[ P = \frac{75 , \text{N}}{0.002375 , \text{m}^2} \approx 31578.95 , \text{Pa} \]
Step 3: Calculate the area of the ram
Now we will calculate the area of the ram: \[ A_{ram} = \pi \left( \frac{d_{ram}}{2} \right)^2 \]
for the ram: \( d_{ram} = 144 , \text{mm} = 0.144 , \text{m} \)
\[ A_{ram} = \pi \left( \frac{0.144}{2} \right)^2 = \pi \left( 0.072 \right)^2 \approx 0.0162585 , \text{m}^2 \]
Step 4: Calculate the force exerted by the ram
Using the pressure calculated previously: \[ F_{ram} = P \cdot A_{ram} \] \[ F_{ram} = 31578.95 , \text{Pa} \cdot 0.0162585 , \text{m}^2 \approx 512.13 , \text{N} \]
So, the force exerted by the ram is approximately: \[ \text{Force exerted by the ram} \approx 512.13 , \text{N} \]
Summary
- Volume of liquid per stroke: \( \approx 251 , cm^3 \)
- Force exerted by the ram: \( \approx 512.13 , N \)