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Chapter 7: Problem 1

Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance \(s\) from the origin is the given function. $$ s=3 \cos 5 t $$

Short Answer

Expert verified
Amplitude: 3, Period: $$ \frac{2\pi}{5} $$, Frequency: $$ \frac{5}{2\pi} $$, Velocity Amplitude: 15.

Step by step solution

01

Identify the given function

The function describing the particle's motion is given by $$ s(t) = 3 \cos 5t $$. This represents a cosine function.
02

Determine the amplitude

The amplitude of the function is the coefficient of the cosine term. For $$ s(t) = 3 \cos 5t $$, the amplitude is 3.
03

Calculate the period

The period of a cosine function of the form $$ s(t) = A \cos(Bt) $$ is given by $$ T = \frac{2\pi}{B} $$. Here, $$ B = 5 $$, so the period is $$ T = \frac{2\pi}{5} $.
04

Determine the frequency

The frequency is the reciprocal of the period. Thus, $$ \text{Frequency} = \frac{1}{T} => \frac{1}{\frac{2\pi}{5}} = \frac{5}{2\pi} $$. Therefore, the frequency is $$ \frac{5}{2\pi} $$.
05

Find the velocity and its amplitude

Velocity is the derivative of the position function with respect to time: $$ v(t) = \frac{ds}{dt} = \frac{d}{dt}[3 \cos 5t] $$. Using the chain rule, $$ v(t) = 3 \cdot (-5 \sin 5t) = -15 \sin 5t $$. The amplitude of velocity is the coefficient of the sine term, which is 15.

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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, the amplitude represents the maximum displacement of the particle from its equilibrium position (origin). Think of it as how far the particle swings away from the middle point in either direction. For our function, given by: \[ s(t) = 3 \, \cos(5t) \]The amplitude is simply the coefficient in front of the cosine function. Here, that coefficient is 3. So, the amplitude is 3 units. This tells us that our particle moves 3 units to the left and right of the origin during its motion.
period
The period of a function in simple harmonic motion is the time it takes for the particle to complete one full cycle of motion. The function provided is: \[ s(t) = 3 \, \cos(5t) \]For a cosine function of the form \( A \, \cos(Bt) \), the period \( T \) can be calculated using the formula: \[ T = \frac{2\pi}{B} \]Here, \( B \) is 5, thus: \[ T = \frac{2\pi}{5} \]This means it takes the particle \( \frac{2\pi}{5} \) units of time to go through one complete cycle of its motion.
frequency
Frequency indicates how often the particle completes a cycle of motion in a unit of time. In other words, it is the rate of oscillation. Given the period \( T \), frequency \( f \) is the reciprocal of the period: \[ f = \frac{1}{T} \]For our function, the period \( T \) was found to be \( \frac{2\pi}{5} \), so: \[ f = \frac{1}{T} = \frac{1}{\frac{2\pi}{5}} = \frac{5}{2\pi} \]This means the particle completes \( \frac{5}{2\pi} \) cycles per unit of time. Frequency helps us understand how frequently the particle oscillates.
velocity amplitude
Velocity amplitude refers to the maximum speed the particle reaches during its motion. To determine this, we need the derivative of the position function, which is the velocity function. Starting with: \[ s(t) = 3 \, \cos(5t) \]We take the derivative with respect to \( t \): \[ v(t) = \frac{ds}{dt} = \frac{d}{dt}[3 \, \cos(5t)] \]Applying the chain rule, we get: \[ v(t) = 3 \, \cdot (-5 \, \sin(5t)) = -15 \, \sin(5t) \]The amplitude of velocity is the coefficient in front of the sine function, which here is 15. So, the velocity amplitude is 15 units. This represents the highest speed the particle achieves, either in the positive or negative direction.

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Most popular questions from this chapter

A periodic amplitude modulated (AM) radio signal has the form $$ y=(A+B \sin 2 \pi f t) \sin 2 \pi f_{c}\left(t-\frac{x}{v}\right) $$ The factor \(\sin 2 \pi f_{c}(t-x / v)\) is called the carrier wave; it has a very high frequency (called radio frequency; \(f_{\epsilon}\) is of the order of \(10^{6}\) cycles per second). The amplitude of the carrier wave is \((A+B \sin 2 \pi f t)\). This amplitude varies with time-hence the term "amplitude modulation"..-with the much smaller frequency of the sound being transmitted (called audio frequency; \(f\) is of the order of \(10^{2}\) cycles per second). In order to see the general appearance of such a wave, use the following simple but unrealistic data to sketch a graph of \(y\) as a function of \(t\) for \(x=0\) over one period of the amplitude function: \(A=3, B=1, f=1\), \(f_{c}=20 .\) Using trigonometric formulas, show that \(y\) can be written as a sum of three waves of frequencies \(f_{c}, f_{c}+f\), and \(f_{c}-f ;\) the first of these is the carrier wave and the other two are called side bands.

Find the amplitude, period, frequency, wave velocity, and wavelength of the given wave, and sketch it as a function of \(x\) for each of the given values of \(t\), and as a function of \(t\) for each given \(x\). $$ y=3 \sin \pi\left(x-\frac{1}{2} t\right) ; \quad t=0,1,2 ; \quad x=0,1,2 $$

Write an equation for a sinusoidal radio wave of amplitude 10 and frequency 600 kilohertz. Hint : The velocity of a radio wave is the velocity of light, \(c=3\). \(10^{8} \mathrm{~m} / \mathrm{sec}\).

In Problems \(5.1\) to \(5.8\), define each function by the formulas given but on the interval \((-l, l)\). [That is, replace \(\pm \pi\) by \(\pm l\) and \(\pm \pi / 2\) by \(\pm 1 / 2\).) Expand each function in a sine-cosine Fourier series and in a complex exponential Fourier series.

Draw a graph of \(\sin 2 x+\sin 2(x+\pi / 3)\). Hint : Use a trigonometry formula to write this as a single harmonic. What are the period and amplitude?
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