The cork in a bottle is 2.00 cm in diameter. If the pressure inside the bottle is 3.00 atm, then the net force on the cork due to the pressure difference is
A) 37.5 N.
B) 45.5 N.
C) 48.9 N.
D) 55.3 N.
E) 63.6 N.

Question

Quizplus · Accepted Answer

The net force on the cork due to the pressure difference can be calculated using the formula $ F = P \times A $, where $ F $ is the force, $ P $ is the pressure difference, and $ A $ is the area of the circle (in this case, the end of the cork). The pressure difference is 3.00 atm, which needs to be converted to Pascals (the SI unit for pressure), knowing that 1 atm = 101325 Pa. Therefore, $ P = 3.00 \times 101325 \, \text{Pa} $.The area $ A $ of the circle (the end of the cork) is given by $ \pi r^2 $, where $ r $ is the radius of the circle. The diameter is given as 2.00 cm, so the radius is half of that, 1.00 cm, or 0.01 m (since we need to work in SI units).Thus, $ A = \pi \times (0.01 \, \text{m})^2 = \pi \times 0.0001 \, \text{m}^2 $.Substituting the values into the formula gives $ F = 3.00 \times 101325 \, \text{Pa} \times \pi \times 0.0001 \, \text{m}^2 $.Calculating this gives $ F \approx 954 \, \text{N/m}^2 \times \pi \times 0.0001 \, \text{m}^2 \approx 30 \, \text{N} $. However, this calculation seems to have gone awry in the explanation process. Let's correct the calculation:$ F = P \times A = 3.00 \times 101325 \, \text{Pa} \times \pi \times (0.01)^2 \, \text{m}^2 $.$ F = 3 \times 101325 \times \pi \times 0.0001 $.$ F = 301975 \times \pi \times 0.0001 $.$ F \approx 301975 \times 0.000314 $.$ F \approx 94.9 \, \text{N} $.This calculation still seems incorrect based on the options provided, indicating a mistake in the recalculation process. The correct approach to find the force is indeed to multiply the pressure (converted to Pascals) by the area, but the error in calculation needs correction. The correct calculation should directly lead to one of the provided options without the need for conversion errors. Given the discrepancy in the explanation, let's correct the approach:The correct formula and approach is indeed $ F = P \times A $, but the correct calculation should directly align with the options provided, suggesting a mistake in the recalculation or explanation process. The correct calculation involves accurately converting the pressure to Pascals, calculating the area with the correct radius in meters, and applying these values correctly. The options suggest specific numerical values that should result from correctly applying these principles, indicating a precise calculation that matches the physical scenario described.

The Cork in a Bottle Is 2