In Exercises 21–28, an object moves in simple harmonic motion ...

Question

In Exercises 21–28, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, find the following: a. the maximum displacement b. the frequency c. the time required for one cycle. d = 10 cos 2πt

Accepted Answer

Hello, today we're gonna be solving the following problem. So we are told that the following equation describes the periodic motion of a particle for which the unit of H is in meters. And the unit of T is in minutes we want to solve for the amplitude, the frequency and the period of the following equation. So the equation given to us is H is equal to eight cosine of four pi T. Let's go ahead and start by solving for the function's amplitude. Now the amplitude represents the deviation of the particle from the center of the function. And that means that the amplitude represents an absolute measurement. What this means is that the amplitude is going to be the absolute value of the leading coefficient of the cosine graph. The leading coefficient is going to be eight in this case. And that means the amplitude is going to be defined as the absolute value of eight which is going to give us eight. That means the amplitude of the particle is going to be eight m long. Next, let's solve for the period of the function. Now the period for a cosine function is defined as two pi over B and B is the leading coefficient in front of the T term in this case. So the leading coefficient is four pi which means we can rep replace B with four pi. That means the period is going to be two pie over four pi pi can be reduced to just one and 2/4 can be reduced to 1/2. That means the period is given to us as 1/2. And with that being said, the period for the particle is gonna be 1/2 minutes. Now that we have the period of the particle, let's solve for the frequency of the particle. The frequency is defined as one over the period for a cosine function and recall that the period is 1/2. So we can represent the frequency as 1/1 over two. This is in the form of a complex fraction. And in order to simplify this, we're gonna multiply the denominator by its reciprocal and multiply the numerator by the reciprocal the denominator, the denominator is going to simplify to just one leaving us in the numerator with one times 2/1, which is going to simplify to just two. And that means that the frequency of the particle is going to be two cycles per minute. So just to summarize the amplitude of the particle is going to be eight m long, the period is going to be a half minute long and the frequency is going to be two cycles per minute. With that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 21–28, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, find the following: a. the maximum displacement b. the frequency c. the time required for one cycle. d = 10 cos 2πt

Key Concepts

Simple Harmonic Motion (SHM)

Amplitude and Maximum Displacement

Frequency and Period of Oscillation

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