A fan at a rock concert is 30 m from the edge of the stage. The sound intensity level at the edge of the stage is 110 dB. The fan's eardrums have a diameter of 8.4mm.
How much energy is transferred to each eardrum in one second? (Hint: It may help to asume that the speaker is on the stage located 1.0 m from the edge.)answer in J.
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I= I0*10^(L(i)/10) so..
I(stage)= 10^-12*10^(110dB/10)=.1
I(2)/I(1)=r(1)^2/r(2)^2
I(2)/.1=1^2/30^2, I(2)=.1/(30^2)=1.04*10^-4 W/m^2
Area of the ear= piD^2/4= 3.14* .0084^2/4=55.4e-9m^2
P=I*A= 1.04e-4*55.4e-9= 6.16e-9
E=P*t = 6.16e-9*1= 6.16e-9
To find the amount of energy transferred to each eardrum in one second, we can make use of the sound intensity level and the area of the eardrum.
First, let's convert the sound intensity level from decibels (dB) to watts per square meter (W/m²). We can use the formula:
I = Io * 10^(L/10),
where I is the sound intensity in watts per square meter, Io is the reference sound intensity (which is 1.0 x 10^(-12) W/m²), and L is the sound intensity level in decibels.
Given that the sound intensity level is 110 dB, we can plug these values into the equation to find the sound intensity:
I = (1.0 x 10^(-12) W/m²) * 10^(110/10)
= (1.0 x 10^(-12) W/m²) * 10^(11)
= 1.0 x 10^(-12 + 11) W/m²
= 1.0 x 10^(-1) W/m²
= 0.1 W/m²
Next, let's find the area of the eardrum. We are given the diameter, so we can calculate the radius as half the diameter:
radius = 8.4mm / 2
= 4.2mm
= 0.0042m
The area of a circle is given by the formula A = πr²:
A = π * (0.0042m)².
We can use the value of π as approximately 3.14:
A ≈ 3.14 * (0.0042m)²
≈ 3.14 * (0.0042m * 0.0042m)
≈ 3.14 * 0.01764m²
≈ 0.0552856m².
Now, let's calculate the energy transferred to each eardrum in one second. The energy is given by the product of the intensity (I) and the area (A). Since intensity is power per unit area, multiplying it by the area gives us the total power:
Energy = I * A
= (0.1 W/m²) * 0.0552856m²
≈ 0.00552856 J.
Therefore, the amount of energy transferred to each eardrum in one second is approximately 0.00552856 Joules (J).