The position of a 50 g oscillating mass is given by 𝓍(t) = (2.0 cm) cos (10 t ─ π/4), where t is in s. Determine: e. The initial conditions.

Video transcript

Hey everyone. So this problem is working with simple harmonic motion. Let's see what it's asking is a 100 grand mass oscillates with a displacement given by the equation X of T equals three centimeters multiplied by the sign of five T plus pi divided by six where T is in settings find the initial position and velocity of the mets. Our multiple choice answers here are a initial position is 1.3 centimeters and initial velocity is 15 centimeters per second. B initial position is 1.5 centimeters and initial velocity is 13 centimeters per second. C initial position is 2.5 centimeters and initial velocity is 11 centimeters per second or D. Initial position is 1.1 centimeters. Velocity is 25 centimeters per second. OK. So this is a pretty straightforward problem as long as we can recall the relationship between position and velocity, namely that velocity is the derivative of position. So they're asking for the position and velocity, they give us the position. So part A or the first part of this problem is very straightforward. The initial position or position at time equals zero seconds is going to be three centimeters multiplied by the sign of so five T or five multiply by zero is zero. So it's just the sign of pi divided by six. And so that X knot or X of zero seconds is 1.5 centers. And so when we look at our multiple choice answers, we can actually eliminate everything except answer choice B but let's still solve this out to make sure that we are on the right track. OK. So as I mentioned, the relationship between the position and velocity is the key to solving this problem here for the second part. So the derivative of our position is going to be our velocity function. When we take the derivative of this equation, we have three centimeters. Oops, sorry, not that equation. This equation given to us in the problem. So we get three centimeters multiplied by five, multiplied by the cosine of five T plus pi divided by six. And so V dot Or V at time equals zero seconds, we'll plug that in for T and we'll get 15 centimeters multiplied by the cosine of pi over six. Let me plug that into our calculator. We get 13 centimeters per second. And so that is the final answer for this problem. And when we go back and look at answer choice B, we can see that yes part B that aligns. So B is the correct answer here. All right. So that's all we have for you on this one. We'll see you in the next video.