Question

A fan at a rock concert is 50.0 m from the stage, and at this point the sound intensity level is 114 dB. Sound is detected when a sound wave causes the tympanic membrane (the eardrum) to vibrate . Typically, the diameter of this membrane is about 8.40 mm in humans.

A.How much energy is transferred to her eardrums each second ?B.How fast would a 2.00 mg mosquito have to fly to have this much kinetic energy?C.How fast a typical 2.0 mg mosquito would have to fly (in mm/s) to have an amount of energy delivered to the eardrum each second when someone whispers (20 dB) a secret in your ear?D.Compare the mosquito's speeds found in parts (B) and (C).

Answer 1

A)
`P=13.92\ J.s^(-1)`

B)
`v=3730.9912\ m.s^(-1)`

C)
`v=74.44\ mm.s^(-1)`

D) mosquitoes speed in part B is very much larger than that of part C.

Step-by-step explanation:

Given:

• Distance form the sound source,
  `s=50\ m`

• sound intensity level at the given location,
  `\beta=114\ dB`

• diameter of the eardrum membrane in humans,
  `d=8.4 * 10^(-3)\ m`

• We have the minimum detectable intensity to the human ears,
  `I_0=10^(-12)\ W.m^(-2)`

(A)

Now the intensity of the sound at the given location is related mathematically as:

`\beta=10\ log((I)/(I_0) )` ..........................................(1)

`114=10\ log\ ((I)/(10^(-12)) )`

`11.4=log\ I+12\ log\ 10`

`I=0.2512\ W.m^(-2)`

As we know :

`I=(P)/(A)`

`0.2512=(P)/(\pi* (8.4^2)/(4) )`

`P=13.92\ J.s^(-1)` is the energy transferred to the eardrums per second.

(B)

mass of mosquito,
`m=2* 10^(-6)\ kg`

Now the velocity of mosquito for the same kinetic energy:

`KE=(1)/(2) m.v^2`

`13.92=(1)/(2)* 2* 10^(-6)* v^2`

`v=3730.9912\ m.s^(-1)`

(C)

Given:

• Sound intensity,
  `\beta = 20\ dB`

Using eq. (1)

`20=10\ log\ ((I)/(10^(-12)) )`

`2=log\ I+12\ log\ 10`

`I=10^(-10)\ W.m^(-2)`

Now, power:

`P=I.A`

`P=10^(-10)* \pi* (8.4^2)/(4)`

`P=5.54* 10^(-9)\ J.s^(-1)`

Hence:

`KE=(1)/(2) m.v^2`

`5.54* 10^(-9)=0.5* 2* 10^(-6)* v^2`

`v=0.07444\ m.s^(-1)`

`v=74.44\ mm.s^(-1)`

(D)

mosquitoes speed in part B is very much larger than that of part C.