During a plane flight, a passenger experiences ear pain due toa head cold that has clogged his Eustachian tubes. Assumingthe pressure in his tubes remained at 1.00 atm (from sea level) andthe cabin pressure is maintained at 0.900 atm, determine the airpressure force (including its direction) on one eardrum, assuming it has a diameter of 0.800 cm.
d=0.008cm r=0.004cm
p tubes= 1.00 atm
p cabin= 0.900 atm
force=pressure difference*Area
area= PI*.004^2 (note: change pressures to Pascals (N/m^2).
To determine the air pressure force on one eardrum, we can use the formula for pressure:
Pressure (P) = Force (F) / Area (A)
We know that the pressure in the Eustachian tubes is 1.00 atm, and the cabin pressure is 0.900 atm. We need to find the pressure difference between these two values.
Pressure difference = P tubes - P cabin
= 1.00 atm - 0.900 atm
= 0.100 atm
Next, we need to convert the pressure difference to SI units (Pascals). 1 atm is equal to 101325 Pa.
Pressure difference (in Pascals) = pressure difference (in atm) * 101325 Pa/atm
= 0.100 atm * 101325 Pa/atm
= 10132.5 Pa
Now, we can calculate the force on the eardrum. The formula for force is:
Force = Pressure difference * Area
The area of the eardrum can be calculated using the formula for the area of a circle:
Area = π * radius^2
Given that the diameter of the eardrum is 0.800 cm (or 0.008 m), we can find the radius (r) as follows:
Radius (r) = diameter / 2
= 0.008 m / 2
= 0.004 m
Substituting the values into the formula:
Area = π * (0.004 m)^2
= π * (0.004 m) * (0.004 m)
= 0.00005027 m^2
Finally, we can calculate the force:
Force = Pressure difference * Area
= 10132.5 Pa * 0.00005027 m^2
= 0.5093 N
The air pressure force on one eardrum is approximately 0.5093 Newtons. The direction of this force will be inward because the cabin pressure (0.900 atm) is lower than the pressure in the Eustachian tubes (1.00 atm).
To determine the air pressure force on one eardrum, we can use the formula for pressure force:
Pressure force = Pressure difference * Area
First, let's calculate the pressure difference. The pressure difference is the difference between the pressure inside the Eustachian tubes and the pressure in the cabin. In this case:
Pressure difference = p tubes - p cabin
= 1.00 atm - 0.900 atm
= 0.100 atm
Next, calculate the area of the eardrum. The area of a circle is given by the formula:
Area = π * radius^2
Here, the radius of the eardrum is given as 0.004 cm. Convert it to meters:
radius = 0.004 cm = 0.004 cm * (1 m / 100 cm)
= 0.00004 m
Now, calculate the area using the formula:
Area = π * 0.00004 m^2
With the value of the radius, the area of the eardrum is:
Area = 3.14 * 0.00004 m^2
= 0.0001256 m^2
Finally, calculate the pressure force by multiplying the pressure difference by the area:
Pressure force = 0.100 atm * 0.0001256 m^2
= 0.00001256 atm * m^2
The unit for pressure force is N/m^2, or Pascal (Pa). To convert atm * m^2 to Pa, we need to multiply by the conversion factor:
1 atm = 101325 Pa
Pressure force = 0.00001256 atm * m^2 * 101325 Pa / 1 atm
≈ 1.27 Pa
The direction of the pressure force is inward, because the pressure inside the Eustachian tubes is higher than the cabin pressure, so it pushes inwards on the eardrum.
Therefore, the air pressure force on one eardrum, assuming it has a diameter of 0.800 cm, is approximately 1.27 Pascal (Pa), directed inward.