An 8 hour exposure to a sound level of 95.0 dB may cause hearing damage. What energy in joules falls on a 0.850 cm diameter eardrum so exposed?
To calculate the energy that falls on the eardrum, we need to use the formula:
Energy = Power × Time
First, let's calculate the power using the sound level in decibels (dB). The formula to convert sound level to power is:
Power (in watts) = 10^(dB/10)
Given that the sound level is 95.0 dB, we can calculate the power:
Power = 10^(95.0/10)
Next, we need to calculate the time in seconds. We are given that the exposure time is 8 hours, so we need to convert that to seconds:
Time = 8 hours × 60 minutes/hour × 60 seconds/minute
Now that we have the power and time, we can calculate the energy in joules:
Energy = Power × Time
However, before we proceed, we need to convert the eardrum diameter from centimeters to meters since the power unit used for the conversion is in watts (Joules/second) which is in the SI unit system.
Eardrum diameter (in meters) = 0.850 cm × 0.01 m/cm
Now we can calculate the energy:
Energy = (Power × Time) / (Area × Time)
Where Area = π × (Eardrum diameter/2)^2
Substituting the values, we can calculate the energy in joules that falls on the eardrum so exposed.
To calculate the energy falling on the eardrum, we need to use the formula for sound energy.
The formula for sound energy is:
Energy = Power × Time
Given:
Sound Level = 95.0 dB
Exposure Time = 8 hours
Diameter of Eardrum = 0.850 cm
First, let's convert the sound level from decibels (dB) to watts per square meter (W/m^2).
Sound Level in W/m^2 = 10^(Sound Level in dB/10)
Sound Level in W/m^2 = 10^(95/10)
Sound Level in W/m^2 = 10^9.5
Sound Level in W/m^2 = 3.162 x 10^9 W/m^2 (approximately)
Next, let's calculate the power incident on the eardrum.
Power = π × (radius)^2 × Sound Level in W/m^2
First, we need to convert the diameter of the eardrum to radius:
Radius = Diameter / 2
Radius = 0.850 cm / 2
Radius = 0.425 cm
Now, let's convert the radius to meters:
Radius = 0.425 cm / 100
Radius = 0.00425 m
Now, let's calculate the power incident on the eardrum:
Power = π × (0.00425 m)^2 × 3.162 x 10^9 W/m^2
Power = 3.1416 × (0.00425 m)^2 × 3.162 x 10^9 W/m^2
Power = 3.1416 × (0.00425 m)^2 × 3.162 x 10^9 W/m^2
Power = 1.793 x 10^-6 W
Finally, let's calculate the energy falling on the eardrum:
Energy = Power × Time
Energy = 1.793 x 10^-6 W × (8 hours × 3600 seconds/hour)
Energy = 1.793 x 10^-6 W × 28800 seconds
Energy = 0.0517 joules (approximately)
Therefore, the energy that falls on a 0.850 cm diameter eardrum exposed to an 8-hour sound level of 95.0 dB is approximately 0.0517 joules.