Question

the tympanic membrane, or eardrum, is a structure that separates the external and middle parts of the ear (see the figure). it is sensitive to and vibrates in response to changes in air pressure and transmits these vibrations to other structures in the inner ear that lead to the sensation of hearing. under normal conditions, the pressure on the inside and outside of the tympanic membrane are kept approximately equal. the auditory tube, also called the eustachian tube, is responsible for this equilibration. however, rapid changes in external pressure can cause large pressure differentials on the tympanic membrane, causing it to rupture. a differential force across the eardrum membrane as little as 5.0 n can cause a rupture. (a) if the cross-sectional area of the membrane is 1.0 cm2, what is the maximum tolerable pressure difference between the external and inner ear? (b) based on your answer in part (a), to what maximum depth could a person dive in fresh water before rupturing an eardrum? experienced divers can actually go much deeper than the answer to (b) by using techniques to equilibrate the pressure on the inside and outside of the membrane as they dive.

Answer 1

The maximum tolerable pressure difference between the external and inner ear is 500,000 Pa. A person can dive to a maximum depth of approximately 51.02 meters in fresh water before rupturing an eardrum.

Step-by-step explanation:

The maximum tolerable pressure difference between the external and inner ear can be calculated using the formula:

Pressure difference = Force / Area

Given that the force required to rupture the eardrum is 5.0 N and the cross-sectional area of the membrane is 1.0 cm2, we can substitute these values into the formula:

Pressure difference = 5.0 N / 1.0 cm2

Converting the area to m2, we get:

Pressure difference = 5.0 N / (1.0 cm2 / 10,000)

Simplifying, we find that the maximum tolerable pressure difference is 500,000 Pa.

To calculate the maximum depth a person can dive before rupturing an eardrum, we can use the concept of pressure and depth in a fluid:

Pressure = ρgh

Where ρ is the density of the fluid (fresh water), g is the acceleration due to gravity, and h is the depth. Rearranging the formula, we get:

h = Pressure / (ρg)

Plugging in the maximum tolerable pressure difference (500,000 Pa) and the density of fresh water (1000 kg/m3), we can calculate the maximum depth:

h = 500,000 Pa / (1000 kg/m3 * 9.8 m/s2)

Simplifying, we find that a person can dive to a maximum depth of approximately 51.02 meters in fresh water before rupturing an eardrum.