BIO. Ear Damage from Diving. If the force on the tympanic membran...

Question

BIO. Ear Damage from Diving. If the force on the tympanic membrane (eardrum) increases by about 1.5 N above the force from atmospheric pressure, the membrane can be damaged. When you go scuba diving in the ocean, below what depth could damage to your eardrum start to occur? The eardrum is typically 8.2 mm in diameter. (Consult Table 12.1.)

Accepted Answer

Hi everyone. Today, we are going to determine the greatest depth that a sensor can actually go without damaging the sensor itself. So the sensor is used to measure the pressure at various depths in the ocean and it has a higher limit of 2.5 Newton above the force it experiences at atmospheric pressure. So in this case, our 2.5 Newton is gonna be our delta force. So the delta force that the pressure can experience on top of the force at the atmosphere is 2.5 Newton and the diameter of the sensor is 6.4 millimeter which in si unit is going to be 6.4 times 10 to the power of minus 3 m. OK. Recall that force and pressure are related in this for using this formula right here, pressure is equals to the force divided by the cross sectional area that the force is exerting its force to and the pressure underneath a surface of water or anything else can be calculated using the P A PM, the atmospheric pressure plus H row G, which is the effect of the pressure being underneath that surface level. OK. So, we are required to determine the extra pressure to correlate it with the extra force that is going to be able to be handled here because we have delta F, we need delta P. So delta P is essentially going to be pressure minus P ATM because we are correlating this or looking at the force is we are comparing it with the force at which is experienced at the atmosphere. So delta P equals to P minus P ATM. And this P can be substituted using this formula right here. So it's going to be P ATM plus H row G minus P ATM. So the P ATM can cancel out and the delta pressure is essentially just H row G. So we know that this extra pressure actually creates the additional force of 2.5 Newton on an area of our diameter here. OK. So we know that force will equals to P multiplied by a using this just rearranging this formula right here. And delta force is then going to be F minus F initial, which is going to be B multiplied by A minus the P A PM, which is the initial one multiplied by A as well. Taking the ace out, we can combine the pressure, this will be P minus P ATM multiplied by the area. And recall that B minus P A PM, we have calculated earlier which is delta P which is H row G. So delta F is essentially just delta P multiplied by A and delta F is essentially H row G multiplied by A. OK. So now we can actually start solving what our cross sectional area is going to be. So we know the diameter is 6.4 times 10 to the part of minus 3 m. We're assuming that it is going to be a circle. So area is going to be pi R squared, which is going to be PD over two squared. And that will be pi multiplied by the D is 6.4 times 10 to the power of minus three divided by two squared. And this will actually be 3.217 times 10 to the power of minus 5 m squared. OK. We know what the area is. So we pretty much know everything in this uh formula right here. We can substitute that. So delta F is equals to H row G multiplied by a, what we want to know is the depth. So the H the height. So we know what the delta F is. It just 2.5 Newton equals H. The row is going to be the row of the sea which in this case, row of sea water is going to be 10 30 kg per meter cube, OK? 10 30 kg per meter cube. And uh G is going to be 9.8 m per second squared. And the area is just the one that we just calculated just like. So, so rearranging everything we can find what the H is and the H is then going to be 7.7 meter. That will be our answer 7.7 m. So 7.7 m is going to correlate with option B and that will be all for this particular problem. If you guys have any sort of confusion, make sure to check out our other lesson videos. We have a lot of other very similar videos on very similar topics that you guys should check out and yeah, that'll be all for this particular practice. Thank you.

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