A glass bottle of soda is sealed with a screw cap . The absolute pressure of the carbon dioxide inside the bottle is `1.80 xx 10^(5)` Pa. Assuming that the top and bottom surfaces of the cap each have an area of `4.10 xx 10^(-4) m^(2)` , obtain the magnitude of the force that the screw thread exerts on the cap in order to keep it on the bottle . The air pressure outside the bottle is one atmosphere .

Options

A. 18 N
B. 32 N
C. 26 N
D. 58 N

Answer

To solve the problem, we need to calculate the force exerted by the screw thread on the cap of the soda bottle. The force is determined by the difference in pressure between the inside and outside of the bottle, multiplied by the area of the cap.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Absolute pressure inside the bottle (P_in) = \(1.80 \times 10^5 \, \text{Pa}\)
- Atmospheric pressure outside the bottle (P_out) = \(1 \, \text{atm} = 1.013 \times 10^5 \, \text{Pa}\)
- Area of the cap (A) = \(4.10 \times 10^{-4} \, \text{m}^2\)

2. **Calculate the Pressure Difference**:
The net pressure difference (ΔP) acting on the cap can be calculated as:
\[ \Delta P = P_{in} - P_{out} \]
Substituting the values:
\[ \Delta P = (1.80 \times 10^5 \, \text{Pa}) - (1.013 \times 10^5 \, \text{Pa}) = 0.787 \times 10^5 \, \text{Pa} \]

3. **Calculate the Force Exerted by the Pressure Difference**:
The force (F) exerted by the pressure difference on the cap can be calculated using the formula:
\[ F = \Delta P \times A \]
Substituting the values:
\[ F = (0.787 \times 10^5 \, \text{Pa}) \times (4.10 \times 10^{-4} \, \text{m}^2) \]

4. **Perform the Calculation**:
\[ F = 0.787 \times 10^5 \times 4.10 \times 10^{-4} = 3.23 \, \text{N} \]

5. **Final Result**:
The magnitude of the force that the screw thread exerts on the cap to keep it on the bottle is approximately:
\[ F \approx 3.23 \, \text{N} \quad \text{(or about 32 N when rounded)} \]