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Chapter 7: Problem 18

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is at its maximum at time \(t=0\) . amplitude 6.25 in, frequency 60 \(\mathrm{Hz}\)

Short Answer

Expert verified
The function is \( y(t) = 6.25 \cos(120\pi t) \).

Step by step solution

01

Identify the Form of the Function

Simple harmonic motion can be modeled by the function \( y(t) = A \cos(2\pi f t) \) or \( y(t) = A \sin(2\pi f t) \). Since the displacement is at its maximum at \( t=0 \), we use \( y(t) = A \cos(2\pi f t) \).
02

Define the Amplitude

The amplitude \( A \) is the maximum displacement from the rest position. Here, \( A = 6.25 \) inches.
03

Determine the Angular Frequency

Frequency \( f \) is given as 60 Hz, and angular frequency \( \omega \) is related as \( \omega = 2\pi f \). Therefore, \( \omega = 2\pi \times 60 = 120\pi \).
04

Substitute Values into the Function

Substituting \( A = 6.25 \) and \( \omega = 120\pi \) into the cosine function gives us the model for the motion: \( y(t) = 6.25 \cos(120\pi t) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
In simple harmonic motion, amplitude is a crucial measurement. It represents the maximum displacement from the object's equilibrium or rest position. This distance measures how far the object moves from its center spot during vibration or oscillation.
For example, in the exercise provided, the amplitude is 6.25 inches. This means that at its peak, the object moves 6.25 inches away from its rest position.
Didn't get it? Imagine pushing a swing. The height it reaches from its middle point on either side is the amplitude.
  • Amplitude indicates energy: Larger amplitudes mean more energy is needed to create the displacement.
  • It is usually measured from the center, not from one extreme point to another.
  • Amplitude remains constant as long as there is no damping.
Knowing the amplitude helps in determining the intensity and reach of the harmonic motion. It's a key factor to calculate when modeling these systems.
Frequency
Frequency refers to how often an object completes a full cycle of motion in one second. It is measured in hertz (Hz). When the frequency is given, you know how many oscillations occur per second.
In our problem, the frequency is 60 Hz, which means the object completes 60 cycles every second.
  • Frequency determines the speed of oscillation: Higher frequency means the object moves quicker.
  • It is expressed in cycles per second.
  • Frequency is essential in defining the periods of waves and vibrations.
Understanding frequency is vital for analyzing the performance of oscillating systems. Knowing the frequency helps anticipate how fast changes in position occur over time.
Angular Frequency
Angular frequency, denoted by \(\omega\), is related to the frequency, but it describes how quickly the angle changes in radians per second. This is especially useful in circular or oscillating motions.
In the provided solution, the frequency was 60 Hz, leading to an angular frequency of \(120\pi\) radians per second. The formula connecting them is \( \omega = 2\pi f \).
  • Angular frequency is expressed in radians per second.
  • It simplifies calculations in wave physics and engineering when the oscillations are periodic.
  • Angular frequency gives insight into the rotational velocity in circular motions.
Understanding angular frequency helps visualize the speed of rotation or oscillation in an intuitive manner, often leading to simpler solutions for modeling motion.

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Most popular questions from this chapter

\(71-74\). Find all solutions of the equation that lie in the interval \([0, \pi]\) . State each answer correct to two decimal places. $$ \cos x=x $$

Spring-Mass System The frequency of oscillation of an object suspended on a spring depends on the stiffness \(k\) of the spring (called the spring constant) and the mass \(m\) of the object. If the spring is compressed a distance \(a\) and then allowed to oscillate, its displacement is given by $$ f(t)=a \cos \sqrt{k / m} t $$ (a) A 10 -g mass is suspended from a spring with stiffness \(k=3 .\) If the spring is compressed a distance 5 \(\mathrm{cm}\) and then released, find the equation that describes the oscillation of the spring. (b) Find a general formula for the frequency (in terms of \(k\) and \(m ) .\) (c) How is the frequency affected if the mass is increased? Is the oscillation faster or slower? (d) How is the frequency affected if a stiffer spring is used (larger \(k\) )? Is the oscillation faster or slower?

For an object in simple harmonic motion with amplitude \(a\) and period \(2 \pi / \omega,\) find an equation that models the displacement \(y\) at time \(t\) if (a) \(y=0\) at time \(t=0 : y=\) _________ (b) \(y=a\) at time \(t=0 : y=\) _________

Blood Pressure Each time your heart beats, your blood pressure increases, then decreases as the heart rests between beats. A certain person's blood pressure is modeled by the function $$ p(t)=115+25 \sin (160 \pi t) $$ where \(p(t)\) is the pressure in mmHg at time \(t,\) measured in minutes. (a) Find the amplitude, period, and frequency of \(p .\) (b) Sketch a graph of \(p\) . (c) If a person is exercising, his or her heart beats faster. How does this affect the period and frequency of \(p ?\)

Tides The Bay of Fundy in Nova Scotia has the highest tides in the world. In one 12 -hour period the water starts at mean sea level, rises to 21 \(\mathrm{ft}\) above, drops to 21 \(\mathrm{ft}\) below, then returns to mean sea level. Assuming that the motion of the tides is simple harmonic, find an equation that describes the height of the tide in the Bay of Fundy above mean sea level. Sketch a graph that shows the level of the tides over a 12 -hour period.
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