This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Here's how to solve the problem: Given values: Height of water column, h_w = 15 cm = 0.15 m Density of water, _w = 1000 kg/m^3 Density of mercury, _Hg = 13600 kg/m^3 Acceleration due to gravity, g = 10 m/s^2 (used to match the provided solution for part a) a) What is the gauge pressure at the water-mercury interface? Step 1: Calculate the gauge pressure due to the water column. The gauge pressure at a depth h in a fluid is given by P = g h. P_gauge = _w g h_w P_gauge = (1000 kg/m^3)(10 m/s^2)(0.15 m) P_gauge = 1500 Pa The gauge pressure at the water-mercury interface is 1500 Pa. b) Calculate the vertical distance h from the top of the mercury in the right-hand arm of the tube to the top of the water in the left-hand arm. Step 2: Apply the principle of pressure balance. At the same horizontal level (the water-mercury interface in the left arm), the pressure must be equal. Let 2x be the height of the mercury column in the right arm above this interface. The pressure exerted by the water column in the left arm is equal to the pressure exerted by the mercury column of height 2x in the right arm. _w g h_w = _Hg g (2x) h_w _w = 2x _Hg Step 3: Solve for 2x, the height of the mercury column above the interface. 2x = (h_w _w)/(_Hg) 2x = (0.15 m)(1000 kg/m^3)13600 kg/m^3 2x = (150)/(13600) m 2x ≈ 0.011029 m Step 4: Calculate the vertical distance h. The distance h is the difference between the height of the water column (h_w) and the height of the mercury column (2x) above the water-mercury interface. h = h_w - 2x h = 0.15 m - 0.011029 m h = 0.138971 m Convert to centimeters and round to one decimal place: h ≈ 13.9 cm The vertical distance h is 13.9 cm. Send me the next one 📸