QuestionJune 4, 2025

5. You dive 5 .7 m below the surface of a pool.The water enters your ear and presses on your ear drum. The eardrum is a small membrane with diameter 8 mm (1000mm=1m) (a) Calculate the gauge pressure at this depth. (b) Calculate the absolute pressure at this depth. (c) Calculate the total force due to the water and the air above the pool acting on your eardrum. (Density of chlorinated water is 992kg/m^3)

5. You dive 5 .7 m below the surface of a pool.The water enters your ear and presses on your ear drum. The eardrum is a small membrane with diameter 8 mm (1000mm=1m) (a) Calculate the gauge pressure at this depth. (b) Calculate the absolute pressure at this depth. (c) Calculate the total force due to the water and the air above the pool acting on your eardrum. (Density of chlorinated water is 992kg/m^3)
5. You dive 5 .7 m below the surface of a pool.The water enters your ear and presses on your ear drum. The eardrum is a small membrane with diameter 8 mm (1000mm=1m) (a) Calculate the gauge pressure at this depth. (b) Calculate the absolute pressure at this depth. (c) Calculate the total force due to the water and the air above the pool acting on your eardrum. (Density of chlorinated water is 992kg/m^3)

Solution
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Answer

(a) Gauge Pressure: 55429.944 \, \text{Pa} ### (b) Absolute Pressure: 156754.944 \, \text{Pa} ### (c) Total Force: 7.89 \, \text{N} Explanation 1. Calculate the gauge pressure Gauge pressure is given by P_{\text{gauge}} = \rho g h. Here, \rho = 992 \, \text{kg/m}^3, g = 9.81 \, \text{m/s}^2, and h = 5.7 \, \text{m}. So, P_{\text{gauge}} = 992 \times 9.81 \times 5.7. 2. Calculate the absolute pressure Absolute pressure is the sum of atmospheric pressure and gauge pressure: P_{\text{absolute}} = P_{\text{atm}} + P_{\text{gauge}}. Assume P_{\text{atm}} = 101325 \, \text{Pa} (standard atmospheric pressure). 3. Calculate the total force on the eardrum The area of the eardrum is A = \pi r^2, where r = \frac{8}{2} \times 10^{-3} \, \text{m}. Total force F = P_{\text{absolute}} \times A.

Explanation

1. Calculate the gauge pressure<br /> Gauge pressure is given by $P_{\text{gauge}} = \rho g h$. Here, $\rho = 992 \, \text{kg/m}^3$, $g = 9.81 \, \text{m/s}^2$, and $h = 5.7 \, \text{m}$. So, $P_{\text{gauge}} = 992 \times 9.81 \times 5.7$.<br /><br />2. Calculate the absolute pressure<br /> Absolute pressure is the sum of atmospheric pressure and gauge pressure: $P_{\text{absolute}} = P_{\text{atm}} + P_{\text{gauge}}$. Assume $P_{\text{atm}} = 101325 \, \text{Pa}$ (standard atmospheric pressure).<br /><br />3. Calculate the total force on the eardrum<br /> The area of the eardrum is $A = \pi r^2$, where $r = \frac{8}{2} \times 10^{-3} \, \text{m}$. Total force $F = P_{\text{absolute}} \times A$.
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