Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 2

Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance \(s\) from the origin is the given function. $$s=2 \sin (4 t-1)$$

Short Answer

Expert verified
Amplitude = 2, Period = \( \frac{\pi}{2} \), Frequency = \( \frac{2}{\pi} \), Velocity Amplitude = 8

Step by step solution

01

Identify the form of the sine function

The given function is of the form \[ s = A \, \sin(Bt + C) \]By comparing, we find:\[ A = 2 \]\[ B = 4 \]\[ C = -1 \]
02

Find the amplitude

Amplitude is the coefficient of the sine function, which is: \[ A = 2 \]
03

Calculate the period

The period of a sine function is given by:\[ \text{Period} = \frac{2 \pi}{B} \]Substituting the value of \( B \): \[ \text{Period} = \frac{2 \pi}{4} = \frac{\pi}{2} \]
04

Determine the frequency

Frequency is the reciprocal of the period:\[ \text{Frequency} = \frac{1}{\text{Period}} = \frac{1}{\pi/2} = \frac{2}{\pi} \]
05

Calculate the velocity amplitude

Velocity is the derivative of displacement with respect to time. To find this, first find the derivative of \[ s = 2 \, \sin(4t - 1) \]: \[ v = \frac{ds}{dt} = 2 \, \cdot 4 \, \cos(4t - 1) = 8 \cos(4t - 1) \]The amplitude of velocity is the coefficient of the cosine function:\[ \text{Velocity Amplitude} = 8 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

amplitude
In harmonic motion, amplitude refers to the maximum distance a particle moves from its equilibrium position. In the given function, the amplitude is represented by the coefficient of the sine function. For the function: $$s = 2 \, \sin(4t - 1)$$ The amplitude is 2. This tells us that the particle moves 2 units away from the origin at its maximum displacement.
period
The period of harmonic motion is the time it takes for the particle to complete one full cycle of motion. For a sine function like $$s = A \, \sin(Bt + C)$$ the period is calculated by the formula: $$\text{Period} = \frac{2 \pi}{B}$$ By substituting the value of B from our function: $$\text{Period} = \frac{2 \pi}{4} = \frac{\pi}{2}$$ Thus, the period of the particle's motion is \(\frac{\pi}{2}\) seconds. This means it takes \(\frac{\pi}{2}\) seconds to complete one full cycle from its starting point and back.
frequency
Frequency tells us how many cycles of motion occur per unit of time. It is the reciprocal of the period. Using our periodic function: \(s = 2 \sin(4t - 1)\), the period was found to be \(\frac{\pi}{2}\) seconds. Using the formula: \[\text{Frequency} = \frac{1}{\text{Period}}\] we find: \[\text{Frequency} = \frac{1}{(\pi / 2)} = \frac{2}{\pi}\] Therefore, the frequency of the motion is \(\frac{2}{\pi}\) cycles per second, indicating approximately how many cycles the particle completes each second.
velocity amplitude
Velocity amplitude is related to the maximum speed the particle reaches during its motion. To find this, we consider the time derivative of the displacement function. For $$s = 2 \sin(4t - 1)$$, the velocity is: $$v = \frac{\mathrm{d}s}{\mathrm{dt}} = 2 \cdot 4 \cos(4t - 1) = 8 \cos(4t - 1)$$. The amplitude of this velocity, or the maximum speed, is the coefficient of the \(\cos(4t - 1)\) term, which is 8. So, the velocity amplitude is 8 units per second, indicating the particle reaches this speed at its fastest point in the cycle.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

You are given a complex function \(z=f(t) .\) In each case, show that a particle whose coordinate is (a) \(x=\operatorname{Re} z,\) (b) \(y=\operatorname{Im} z\) is undergoing simple harmonic motion, and find the amplitude, period, frequency, and velocity amplitude of the motion. $$\quad z=-4 e^{i(2 t+3 \pi)}$$

Consider one arch of \(f(x)=\sin x .\) Show that the average value of \(f(x)\) over the middle third of the arch is twice the average value over the end thirds.

Find the amplitude, period, frequency, wave velocity, and wavelength of the given wave. By computer, plot on the same axes, \(y\) as a function of \(x\) for the given values of \(t,\) and label each graph with its value of \(t .\) Similarly, plot on the same axes, \(y\) as a function of \(t\) for the given values of \(x,\) and label each curve with its value of \(x.\) $$y=\cos 2 \pi\left(x-\frac{1}{4} t\right) ; \quad t=0,1,2,3 ; \quad x=0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}$$

Each of the following functions is given over one period. Sketch several periods of the corresponding periodic function and expand it in an appropriate Fourier series. $$f(x)=\left\\{\begin{array}{ll}x / 2, & 0< x<2 \\\1, & 2< x<3\end{array}\right.$$.

Find the amplitude, period, frequency, wave velocity, and wavelength of the given wave. By computer, plot on the same axes, \(y\) as a function of \(x\) for the given values of \(t,\) and label each graph with its value of \(t .\) Similarly, plot on the same axes, \(y\) as a function of \(t\) for the given values of \(x,\) and label each curve with its value of \(x.\) $$y=2 \sin \frac{2}{3} \pi(x-3 t) ; \quad t=0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} ; \quad x=0,1,2,3$$
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free