Question

A particular person's eardrum is circular, with a diameter of 7.90 mm. How much sound energy (in J) is delivered to an eardrum in one second, at the threshold of human hearing? (The threshold of human hearing is taken to be 1.00 ✕ 10−12 W/m2.)

Answer 1

The sound energy is
`5*10^(-5)\ J`.

Step-by-step explanation:

Given that,

Diameter = 7.90 mm

Suppose the intensity for threshold of pain is 1.00 W/m².

We need to calculate the area

Formula of area

`A=\pi r^2`

Put the value into the formula

`A=\pi*((7.90*10^(-3))/(2))^2`

`A=0.000049`

`A=4.9*10^(-5)\ m^2`

`A=5*10^(-5)\ m^2`

We need to calculate the power

Using formula of power

`P=IA`

Where, P = Power

I = intensity

A = area

Put the value into the formula

`P=1.00*10^(-12)*5*10^(-5)`

`P=5*10^(-17)\ W`

We need to calculate the incident power at the threshold of pain

Using formula of power

`P=IA`

`P=1.00*5*10^(-5)`

`P=5*10^(-5)\ W`

We need to calculate the energy

Using formula of energy

`E=Pt`

Where. P = power

t = time

Put the value into the formula

`E=5*10^(-5)*1`

`E=5*10^(-5)\ J`

Hence, The sound energy is
`5*10^(-5)\ J`.

Answer 2

`4.90167* 10^(-17)\ J`

Step-by-step explanation:

I = Hearing intensity =
`1* 10^(-12)\ W/m^2`

A = Area =
`\pi r^2`

d = Diameter = 7.9 mm

r = Radius =
`(d)/(2)=(7.9)/(2)=3.95\ mm`

Power is given by

`P=IA\\\Rightarrow P=1* 10^(-12)* \pi * (3.95* 10^(-3))^2\\\Rightarrow P=4.90167* 10^(-17)\ W`

t = Time the eardrum is exposed to sound = 1 second

Energy is given by

`E=Pt\\\Rightarrow E=4.90167* 10^(-17)* 1\\\Rightarrow E=4.90167* 10^(-17)\ J`

The energy transferred to the eardrum is
`4.90167* 10^(-17)\ J`