The radius of a press cylinder in a hydraulic press is double the diameter of the pump cylinder. Then

Options

A. mechanical advantage is 16.
B. the work done on the load is equal to the work done by the effort.
C. mechanical advantage is 4
D. Both (a) and (b)

Answer

To solve the problem, we need to determine the mechanical advantage of a hydraulic press where the radius of the press cylinder is double the diameter of the pump cylinder.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let the diameter of the pump cylinder be \( D \).
- Therefore, the radius of the pump cylinder \( r_p \) is given by:
\[ r_p = \frac{D}{2} \]
- The radius of the press cylinder \( r_c \) is double the diameter of the pump cylinder:
\[ r_c = 2D \]

2. **Calculate the Mechanical Advantage (MA):**
- The mechanical advantage of a hydraulic press is defined as the ratio of the area of the press cylinder to the area of the pump cylinder:
\[ MA = \frac{\text{Area of press cylinder}}{\text{Area of pump cylinder}} = \frac{\pi r_c^2}{\pi r_p^2} \]
- Since \(\pi\) cancels out, we have:
\[ MA = \frac{r_c^2}{r_p^2} \]

3. **Substitute the Values:**
- Substitute \( r_c = 2D \) and \( r_p = \frac{D}{2} \):
\[ MA = \frac{(2D)^2}{\left(\frac{D}{2}\right)^2} \]

4. **Simplify the Expression:**
- Calculate \( (2D)^2 \) and \( \left(\frac{D}{2}\right)^2 \):
\[ (2D)^2 = 4D^2 \]
\[ \left(\frac{D}{2}\right)^2 = \frac{D^2}{4} \]
- Now substitute these back into the MA equation:
\[ MA = \frac{4D^2}{\frac{D^2}{4}} = 4D^2 \times \frac{4}{D^2} = 16 \]

5. **Conclusion:**
- The mechanical advantage of the hydraulic press is:
\[ MA = 16 \]

### Final Answer:
The mechanical advantage of the hydraulic press is 16.

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