Vision is blurred if the head is vibrated at 29 Hz because the vibrations are resonant with the natural frequency of the eyeball in its socket. If the mass of the eyeball is 7.5 g, a typical value, what is the effective spring constant of the musculature that holds the eyeball in the socket?

Video transcript

Hey, everyone. So this problem is dealing with simple harmonic motion. Let's submit its a, a person standing on the bridge notices that the bridge starts to vibrate with a frequency of 3.5 Hertz when a large truck passes by the mass of the bridge is 5000 kg. And then we're asked to find the effective spring constant of that bridge. Our multiple choice answers here are a 1.9 times 10 to the three Newton per meter. B 1.9 times 10 to the sixth Newton per meter. C 2.42 times 10 to the three Newton per meter or D 2.42 times 10 to the sixth Newton per meter. And so the key to solving this problem is going to be we're calling our period equation in terms of our spring constant. So that's gonna be T is equal to two pi multiplied by the square root of M divided by K, where T is our period M is our mass and K is that effective spring constant that we're solving for. In turn, we can recall that by definition, a period is just one divided by the frequency. And so when the plug that into this initial equation and square both sides, we're left with one divided by F squared equals four pi squared multiplied by M all divided by K. And again, we're solving for K, we wanna isolate that variable. And that leaves us with four pi squared and four pi squared M multiplied by F swear. And so from there, it's a simple plug-in chug using the information given to us in the problem. So we have K equals four pi squared multiplied by the mass of 5000 kg multiplied by our frequency 3.5 Hertz. And that quantity is squared. And we plug that in to our calculator and we get 2.42 times 10 to the six newtons per meter. And that aligns with answer choice T. So D is the correct answer for this problem. So that's all we have for this one. We'll see you in our next video.