Question

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 7.9 mm in diameter. Take the density of seawater to be equal 1.03×103kg/m3 .

Answer 1

The pressure at a depth h in a fluid with density ρ is given by:

P = ρgh

where g is the acceleration due to gravity.

The force exerted on the eardrum is equal to the pressure difference between the inside and outside of the eardrum multiplied by the area of the eardrum:

F = A(Pinside - Poutside)

where A is the area of the eardrum, Pinside is the pressure inside the ear, and Poutside is the pressure outside the ear.

To find the depth at which the force on the eardrum is 1.5 N above atmospheric pressure, we need to first find the pressure difference and then the corresponding depth.

The atmospheric pressure at sea level is approximately 101,325 Pa. Adding 1.5 N of force per the eardrum means adding 1.5 N of force per the area of the eardrum, which is:

A = π(0.0079 m/2)^2 = 4.909×10^-5 m^2

So the pressure difference is:

ΔP = F/A = 1.5 N/4.909×10^-5 m^2 = 3.057×10^4 Pa

Now we can solve for the depth using the pressure equation:

ΔP = ρgh

h = ΔP/(ρg) = (3.057×10^4 Pa)/(1.03×10^3 kg/m^3 × 9.81 m/s^2) ≈ 3.02 m

Therefore, the eardrum can be damaged when scuba diving at a depth of approximately 3.02 meters in seawater.