Sound is detected when a sound wave causes the tympanic membrane (the eardrum) to vibrate. Typically, the diameter of this membrane is about 8.4 mm in humans. (a) How much energy is delivered to the eardrum each second when someone whispers (20 dB) a secret in your ear?

Video transcript

Hey everyone. So for this video we are going to be working with sound intensity and sound energy. So let's see what the problem is asking us. The microphone has a diaphragm diameter of 12. millimeters. The singer is singing at 90 decibels. So how much energy does that singer supply to the diaphragm? Alright, let's start by writing down our givens. So the diameter is 12.7 millimeters. I'm just gonna take a second to rewrite that as 0.127 m. We also know that the sound intensity level is 90 decibels. So first we need to recall our sound energy equation that's given as E equals I A T. Where I is, the intensity is area and T is time. Were also given um sound intensity level. So let's recall our sound intensity level equation is beta equals 10 decibels times the log of I over I not where I is your intensity and I not is the intensity at the threshold of hearing. That's also a constant. So let's recall that that is one times 10 to the 12th watts per meter squared. Alright, so we have enough to solve, We know enough from the problem and from our equations to solve this second equation for intensity, we're gonna use that to plug it into our energy equation. So you know, 90 decibels equals 10 decibels times the log of I over I not, we're gonna divide each side by 10 decibels. You are left with nine equals the log of I. Over I not, we need to remember some log math here and this can actually be rewritten as I over. I not equals 10 to the nine Will solve her intensity I which is 10 to the nine times 10 to the -12 and that equals 10 to the minus three watts per meter squared. Alright, now let's look at our energy equation. So now we know intensity time is a constant, it's just gonna be instantaneous, energy is just one second. But what we still don't know is area were given diameter. So let's recall that the area of a circle is pi R squared, where are is just Diameter divided by two. Alright, so let's plug our diameter in 0.012, seven m divided by two squared. That gives us an area of 1.26, 7 Times 10 to the -4 square meters. Alright, so now we do have everything that we need to solve our energy equation, I a T. So our intensity is one times 10 to the negative three watts per meter squared. Our area is 1.26, 7 times 10 to the negative four m squared. And our time is just 1/2. That gives us an energy of 1.26, 7 times 10 to the -7 jewels. Take a look at our possible answers and that is d 1.2, 7 times 10 to the -7 jewels. Alright, that's all for this problem. We'll see you in the next video