An object oscillates along a vertical line, and its position i...

Question

An object oscillates along a vertical line, and its position in centimeters is given by y(t)=30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

At what times and positions is the velocity zero?

Accepted Answer

Hello. In this video, we are going to be working on the following velocity problem. We are told that a wave travels along a string and its position in millimeters is described by the function H of T equal to 15 multiplied by the quantity to cosine of T + 1. For T greater than or equal to 0, where T represents time in seconds, we want to determine the times and positions at which the velocity is zero. So, how do we start this process? Well, first, notice that we are working with the velocity, and we are told that the position is described by the function H of T given to us. So since we need velocity, we will need to take the derivative of position in order to find our velocity function. So the position function given to us is H of T equal to 15 multiplied by 2, cosine of T. Plus one Now, before we take the derivative, let's try to simplify the original position function. We can simplify it by distributing the value of 15 across the quantity to cosine of T + 1. That will give us 30 cosine of T. Plus 15. And now that this is simplified, let's go ahead and take the derivative of this position function. By taking the derivative with respect to time, the variable T, the derivative, which we'll call V of T for velocity, is going to be 30 multiplied by the derivative of cosine, which will be negative sine. So we have negative 30, sine of T. And the derivative of a constant value 15 is going to be zero. So this is going to be our velocity function. Now that we have our velocity function, let's start by determining the time when the velocity is zero. So, in order to find the time when velocity is 0, we are going to take our velocity function and set it equal to 0, and solve for T. That is going to give us -30s of T. Equal to 0 If we divide both sides of this equation by -30, we will have sin of T equal to 0. Now, sine of T equal to 0 has multiple solutions. Those solutions are going to be T is equal to 0. Pie and to pie, and so on and so on. So we were going to have to represent this as a multiple of pi. What we can do is we can state that T is equal to N multiplied by pi, where N is a positive integer. This will mean that for any time that is a multiple, a positive multiple of pi, that time is where the velocity is equal to 0, and that is going to be the solution to the first part of this problem. Now that we have the time represented as a variable where velocity is 0, we are going to need to use this time in order to solve for the position when the velocity is 0. In order to solve that part, we are going to take the time we have just solved for. T is equal to N multiplied by pi and plug it in to our position function. That is going to give us H of N pi equal to 30 multiplied by cosine. Of Nie Plus 15. Now there are two total outcomes that can happen when we plug in N pi to our position function. One is that N can be an even integer. Or n can be an odd integer. We are going to have to consider both cases when trying to solve for the positions when the velocity is 0. Let's go ahead and start by assuming that the integer N is positive and even. So if N is even, that is going to give us 30 multiplied by cosine of pi. And when N is even, cosine is going to simplify to be 1. So that is going to leave us with 30 multiplied by 1 plus 15, which is going to simplify to be 45 m. Millimeter. So The position at 45 millimeters is when the velocity will be zero. Now, what about the opposite case when N is odd? If N is odd, then cosine of N i will simplify to be -1. That is going to be 30 multiplied by -1 + 15, which will further simplify to be -15 millimeters. So that means that at the position at -150 millimeters, the velocity is going to be zero. So, in total, we have 2 positions where velocity will be 0, and this is going to be the total solution to this problem. So I hope this video helps you in understanding how to approach this type of problem, and I will go ahead and see you all in the next video.

An object oscillates along a vertical line, and its position in centimeters is given by y(t)=30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.
At what times and positions is the velocity zero?

Key Concepts

Velocity

Critical Points

Oscillation

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