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.)

Verified step by step guidance

1. First, calculate the area of the tympanic membrane (eardrum) using the formula for the area of a circle: \( A = \pi r^2 \). The diameter is given as 8.2 mm, so the radius \( r \) is half of that. Convert the radius from millimeters to meters for consistency in SI units.
2. Next, determine the pressure difference that would cause a force of 1.5 N on the eardrum. Use the formula \( F = P \times A \), where \( F \) is the force, \( P \) is the pressure difference, and \( A \) is the area. Rearrange this formula to solve for the pressure difference: \( P = \frac{F}{A} \).
3. The pressure difference \( P \) is due to the water pressure at a certain depth minus the atmospheric pressure. Water pressure at a depth \( h \) can be calculated using the formula \( P_{water} = \rho g h \), where \( \rho \) is the density of seawater (approximately 1025 kg/m³), \( g \) is the acceleration due to gravity (9.81 m/s²), and \( h \) is the depth in meters.
4. Set the pressure difference \( P \) equal to \( P_{water} - P_{atm} \), where \( P_{atm} \) is the atmospheric pressure (approximately 101,325 Pa). Solve for the depth \( h \) by rearranging the equation: \( h = \frac{P + P_{atm}}{\rho g} \).
5. Substitute the known values into the equation to find the depth \( h \) at which the pressure difference would be sufficient to cause a force of 1.5 N on the eardrum, potentially leading to damage.

Key Concepts

Pressure and Force Relationship

Pressure is defined as force per unit area. When diving, the pressure exerted by the water increases with depth, adding to the atmospheric pressure already present. The force on the eardrum is the product of this pressure and the area of the eardrum. Understanding this relationship is crucial to determine the depth at which the additional force could cause damage.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases linearly with depth, calculated as the product of the fluid's density, gravitational acceleration, and depth. This concept helps determine how much additional pressure is applied to the eardrum as a diver descends underwater.

Area of a Circle

The area of a circle is calculated using the formula A = πr², where r is the radius. For the eardrum, knowing its diameter allows us to find the radius and subsequently the area. This area is essential for calculating the force exerted on the eardrum by the pressure at a given depth, which is necessary to assess the risk of damage.