A chemistry graduate student is designing a pressure vessel for an experiment. The vessel will contain gases at pressures up to 180.0MPa. The student's design calls for an observation port on the side of the vessel (see diagram below). The bolts that hold the cover of this port onto the vessel can safely withstand a force of 3.50MN.Calculate the maximum safe diameter w of the port. Round your answer to the nearest 0.1cm.
Question
A chemistry graduate student is designing a pressure vessel for an experiment. The vessel will contain gases at pressures up to 180.0MPa. The student's design calls for an observation port on the side of the vessel (see diagram below). The bolts that hold the cover of this port onto the vessel can safely withstand a force of 3.50MN.Calculate the maximum safe diameter w of the port. Round your answer to the nearest 0.1cm.
Solution
To solve this problem, we need to use the formula for pressure, which is P = F/A, where P is pressure, F is force, and A is area. We need to solve for the diameter of the port, so we need to rearrange the formula to solve for A, and then convert that to a diameter.
Step 1: Rearrange the formula to solve for A. A = F/P.
Step 2: Substitute the given values into the formula. A = 3.50MN / 180.0MPa = 0.019444 m^2.
Step 3: The area of a circle is given by the formula A = πr^2, where r is the radius. We can rearrange this to solve for r: r = sqrt(A/π).
Step 4: Substitute the value for A into the formula to find r. r = sqrt(0.019444 m^2 / π) = 0.0785 m.
Step 5: The diameter of the circle is twice the radius, so d = 2r. d = 2 * 0.0785 m = 0.157 m.
Step 6: Convert the diameter from meters to centimeters. 1 m = 100 cm, so d = 0.157 m * 100 = 15.7 cm.
So, the maximum safe diameter of the port is approximately 15.7 cm.
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