Two syringes of different cross-section (without needle filled with water are connected with a tightly fitted rubber tube filled with water. Diameters of the smaller piston an larger piston are `1 cm` and `3 cm` respectively. If a force of 10N is applied to the smaller piston then the force exerted on the larger piston is

Options

A. 30N
B. 60 N
C. 90 N
D. 100 N

Explanation

To solve the problem of finding the force exerted on the larger piston when a force is applied to the smaller piston, we can use the principle of Pascal's law, which states that pressure applied to a confined fluid is transmitted undiminished throughout the fluid.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Diameter of the smaller piston, \( d_1 = 1 \, \text{cm} = 0.01 \, \text{m} \)
- Diameter of the larger piston, \( d_2 = 3 \, \text{cm} = 0.03 \, \text{m} \)
- Force applied to the smaller piston, \( F_1 = 10 \, \text{N} \)

2. **Calculate the Radii of the Pistons:**
- Radius of the smaller piston, \( r_1 = \frac{d_1}{2} = \frac{0.01}{2} = 0.005 \, \text{m} \)
- Radius of the larger piston, \( r_2 = \frac{d_2}{2} = \frac{0.03}{2} = 0.015 \, \text{m} \)

3. **Calculate the Areas of the Pistons:**
- Area of the smaller piston, \( A_1 = \pi r_1^2 = \pi (0.005)^2 = \pi \times 0.000025 \approx 7.85 \times 10^{-5} \, \text{m}^2 \)
- Area of the larger piston, \( A_2 = \pi r_2^2 = \pi (0.015)^2 = \pi \times 0.000225 \approx 7.07 \times 10^{-4} \, \text{m}^2 \)

4. **Apply Pascal's Law:**
- According to Pascal's law, the pressure exerted on the smaller piston is equal to the pressure exerted on the larger piston:
\[ \frac{F_1}{A_1} = \frac{F_2}{A_2} \]
Rearranging gives:
\[ F_2 = F_1 \cdot \frac{A_2}{A_1} \]

5. **Substituting the Values:**
- Substitute \( F_1 = 10 \, \text{N} \), \( A_1 \), and \( A_2 \):
\[ F_2 = 10 \cdot \frac{7.07 \times 10^{-4}}{7.85 \times 10^{-5}} \]
- Calculate the ratio:
\[ \frac{A_2}{A_1} = \frac{7.07 \times 10^{-4}}{7.85 \times 10^{-5}} \approx 9 \]
- Therefore:
\[ F_2 = 10 \cdot 9 = 90 \, \text{N} \]

6. **Conclusion:**
- The force exerted on the larger piston is \( F_2 = 90 \, \text{N} \).