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

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 = −8 cos π/2 t

Accepted Answer

Hello, today we're going to 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. So let's take a look at the given equation. The equation given to us is H is equal to negative eight cosine of pi over four T. Let's start by solving for the function's amplitude. So the amplitude represents the deviation of the particle from the center of the equation. That means the amplitude represents a measurement and a measurement can never be negative. The amplitude is going to be the absolute value of the leading coefficient and the leading coefficient in this case is going to be negative eight. So the amplitude is going to be represented as the absolute value of negative eight and that is going to give us a value of eight. What this means is that the amplitude of the particle is going to be eight m long. Now that we have the amplitude, let's go ahead and solve for the period, the period for a cosine function is defined as two pi over B B represents the coefficient in front of the T term in this case. And the coefficient in front of the T term is going to be pi over four. So we can replace B with pi over four. And that is going to give us a period of two pi over pi over four. 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 of the denominator. This is going to simplify the denominator to give us just one. And what we are left with is two pi multiplied by four over pi we can represent two pi as a fraction by putting it over one. And we can reduce the P in the numerator and denominator to just one. This is going to leave us with two times four in the numerator which is just 8/1 times one, which is one. And this is going to simplify to give us eight. What this means is that the period is defined as eight minutes for the particle. Now that we have the period, let's go ahead and solve for the frequency. Now, the frequency of a cosine function is defined as one over the period. And remember that we found the period to be a total of eight minutes. So we can represent the frequency as 1/8. Since this is in simplest form, that means the frequency is going to be 1/8 cycles per minute for the given particle. So just to summarize, we've calculated the amplitude which is eight m, we've calculated the period which is going to be eight minutes. And we calculated the frequency to be 1/8 cycles per minute. With that being said, the answer to this problem is going to be D 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 = −8 cos π/2 t

Key Concepts

Simple Harmonic Motion (SHM)

Maximum Displacement

Frequency and Period

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