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To solve this problem, we need to determine the pressure \( p_2 \) for the two cases specified: 

1. **Given Data:**
   - The hydraulic cylinder has a piston diameter \( D_p = 2.5 \) in and a rod diameter \( D_r = 0.25 \) in.
   - The pressure relief valve setting is 1600 psi.
   - The suspended load \( W \) is 575 lb in part (a), and \( W = 0 \) lb in part (b).

2. **Solution Approach:**
   - We need to find the pressure \( p_2 \), which is the pressure in the chamber of the hydraulic cylinder when moving at a constant speed.
   - The pressure \( p_2 \) can be calculated using the force balance on the piston, considering the hydraulic forces acting on the areas of the piston and the rod.

3. **Hydraulic Cylinder Force Calculation:**
   - The effective area of the piston (\( A_p \)) and the annular area on the rod side (\( A_a \)) can be calculated using the formulas:
     \[
     A_p = \frac{\pi}{4} D_p^2
     \]
     \[
     A_a = \frac{\pi}{4} (D_p^2 - D_r^2)
     \]
   - Then, using the force balance on the piston side, the pressure \( p_2 \) can be found from:
     \[
     p_2 = \frac{W}{A_a}
     \]
   - For part (b), where \( W = 0 \), the pressure \( p_2 \) would theoretically be zero, as there is no load requiring hydraulic pressure to balance.

Let's calculate the areas:

### Step 1: Calculate Areas
- Piston area, \( A_p \):
  \[
  A_p = \frac{\pi}{4} (2.5)^2 = 4.91 \text{ in}^2
  \]
- Annular area, \( A_a \):
  \[
  A_a = \frac{\pi}{4} (2.5^2 - 0.25^2) = 4.90 \text{ in}^2
  \]

### Step 2: Calculate \( p_2 \) for \( W = 575 \text{ lb} \)
- Use:
  \[
  p_2 = \frac{575}{4.90} = 117.35 \text{ psi}
  \]

### Step 3: Case \( W = 0 \)
- When \( W = 0 \), the pressure \( p_2 \) is 0 psi since no load exists.

Would you like more details or have any questions?

### Related Questions:
1. How does the rod diameter affect the calculation of annular area?
2. What happens to \( p_2 \) if the pressure relief valve setting is increased?
3. How would you determine the flow rate through the needle valve?
4. What role does fluid viscosity play in this problem?
5. Can you calculate the pressure difference if the load increases to 1000 lb?

### Tip:
When working with hydraulic systems, always verify the units used in pressure and area calculations to avoid errors.

The system shown below has a hydraulic cylinder with a suspended load W. The cylinder piston and the rod diameters are 2.5 in and 0.25 in respectively. The pressure relief valve setting is 1600 psi. Determine pressure p2 for a constant cylinder speed if (a) W = 575 lb (b) W = 0 (load is removed).

Learn how to calculate the pressure in a hydraulic cylinder with a piston diameter of 2.5 inches and rod diameter of 0.25 inches when supporting a suspended load of 575 lb. Step-by-step explanation of force balance and pressure determination, including both loaded and unloaded cases.

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