The point of the needle of a sewing machine moves in SHM along the x-axis with a frequency of 2.5 Hz. At t = 0 its position and velocity components are +1.1 cm and -15 cm/s, respectively. (a) Find the acceleration component of the needle at t = 0.

Video transcript

Everyone in this problem. The head of a periodic camera moves in simple harmonic motion. A parallel to the X axis at a frequency of eight hertz. When T is equal to 1.58 seconds. The position of the head is negative 2.3 centimeters. Well its velocity is positive 20 centimeters per second. And were asked to determine the acceleration of the head at T equals 1.58 seconds. Alright, so we're talking about acceleration and simple harmonic motion. Let's recall that we have the following. Okay, the acceleration A is equal to negative K. X. K. Where K is the spring constant divided by M. Alright. So do we have everything we need for this equation? Well we have X. Okay, this is the position Which we're told is negative 2.3 cm. But we don't know this negative K Over M. All right now we could try to find K on its own M. On its own and plug into this equation. But let's recall that we actually have the following the angular frequency omega Is equal to two pi in terms of frequency f. And this is equal to the square root of K. Over app. We know the frequency and this is going to allow us to calculate K over M. Okay, so we don't even need to find can its own and I'm on its own. We can just find this ratio K over it. Alright, so if we were to square both sides, we get two pi F squared is equal to K over M. It's okay. Over. M is going to be equal to two pi times the frequency is eight hertz all swear. And this gives us a value for K over M. 256 pi squared. The unit is hurt squared. Okay. Alright. So now we know this ratio of K over M. And we know X. So we can get back to our acceleration equation. So the acceleration is going to be equal to Well, K over M is 256 pi squared. So we have negative 256 pi squared in our unit here is hurts squared times the position X and position X is negative 2.3 centimeters. Okay, we want to write this in terms of meters, this is going to be negative 2.3 centimeters times one m per 100 centimeters. Okay. The unit of centimeter will divide out. We're gonna be left with negative 0. m. Okay, so to convert from centimeters to meters we divide by 100. Alright, so we're gonna multiply by this value negative 0. m. Okay, let's just do a dotted line so we can make sure we separate the two. All right, so we have heard squared which is one over second squared. Ok. And then we have times meters. Our unit will be meters per second squared which is what we want for acceleration and we get an acceleration Of approximately 58.1 m/s squared. Okay. And so our acceleration 58.1 m per second squared. That's gonna be answer a thanks everyone for watching. I hope this video helped see you in the next one.