At a rock concert, the sound intensity 1.0 in front of the bank of loudspeakers is 0.10 . A fan is 30 from the loudspeakers. Her eardrums have a diameter of 8.4 .How much energy is transferred to each eardrum in one second?
The answer is in joules but I have no idea how to get there using the sound intensity formulas :(
To calculate the energy transferred to each eardrum in one second, we need to use the formula for sound intensity:
Sound Intensity (I) = Power (P) / Area (A)
We can rearrange the formula to solve for power:
Power (P) = Sound Intensity (I) x Area (A)
First, let's calculate the area of each eardrum:
Area = π x (radius)²
Given that the diameter of each eardrum is 8.4, the radius will be half of the diameter:
Radius = 8.4 / 2 = 4.2
Substituting the values into the formula:
Area = π x (4.2)²
Now, calculate the area and substitute the value into the power formula:
Power = 0.10 x π x (4.2)²
Finally, calculate the power:
Power = 0.10 x π x 4.2²
The resulting value will give us the energy transferred to each eardrum in one second.
To calculate the energy transferred to each eardrum in one second, we need to use the formula for sound intensity:
Sound Intensity = Power / Area
Given information:
Sound intensity in front of the loudspeakers (I₁) = 0.10 W/m²
Distance from the loudspeakers to the fan (r) = 30 meters
Eardrum diameter (d) = 8.4 meters
First, let's find the area of each eardrum:
Area = π * (radius)²
To find the radius:
Radius = diameter / 2
Radius = 8.4 / 2 = 4.2 meters
Now, we can calculate the area:
Area = π * (4.2)²
Next, rearrange the formula for sound intensity to solve for power:
Power = Sound Intensity * Area
Substituting the given values, we get:
Power = 0.10 * π * (4.2)²
Finally, multiply the power by the time duration (one second) to obtain the energy transferred to each eardrum in one second.
The sound travels in all directions .. so it's power is spread out over the surface of a sphere.
The radius of this sphere increases as it travels, distributing the power (P) over a greater and greater area (A) that reduces it's intensity (I) in W/m²
Intensity, I = P/A .. for a given source power the intensity I ∝ 1/A ∝ 1/d²
I ∝ 1/d² ..
I1 / I2 = (d2)² / (d1)²
Intensity at 30m .. I2 = I1 x (d1)² / (d2)² ..
I2 = 0.10W/m² x (1.0m)² / (30.0m)² .. .. I2 = 1.10^-4 (J/s)/m² .. W/m²
Energy collected by ear drum each sec .. E = I2 x Ear drum area
E = 1.10^-4 (J/s)/m² x (π.[4.20^-3m]²) .. .. .. ►E = 6.10^-9 J