Copyright © 2018 John Wiley & Sons, Inc. and primarily advanced by Prof. A. Iskandar.
15-1 Simple Harmonic Motion (1 of 20) Link to heading
Learning Objectives
- 15.01 Distinguish simple harmonic motion from other types of periodic motion.
- 15.02 For a simple harmonic oscillator, apply the relationship between position $x$ and time $t$ to calculate either if given a value for the other.
- 15.03 Relate period $T$, frequency $f$, and angular frequency $\omega$.
15-1 Simple Harmonic Motion (2 of 20) Link to heading
- 15.04 Identify (displacement) amplitude $x_m$, phase constant (or phase angle) $\phi$, and phase $\omega t + \phi$.
- 15.05 Sketch a graph of the oscillator’s position $x$ versus time $t$, identifying amplitude $x_m$ and period $T$.
- 15.06 From a graph of position versus time, velocity versus time, or acceleration versus time, determine the amplitude of the plot and the value of the phase constant $\phi$.
15-1 Simple Harmonic Motion (3 of 20) Link to heading
- 15.07 On a graph of position $x$ versus time $t$ describe the effects of changing period $T$, frequency $f$, amplitude
- 15.08 Identify the phase constant $x_m$, or phase constant $\phi$ that corresponds to the starting time ($t = 0$) being set when a particle in SHM is at an extreme point or passing through the center point.
15-1 Simple Harmonic Motion (4 of 20) Link to heading
- 15.09 Given an oscillator’s position $x(t)$ as a function of time, find its velocity $v(t)$ as a function of time, identify the velocity amplitude vm in the result, and calculate the velocity at any given time.
- 15.10 Sketch a graph of an oscillator’s velocity $v$ versus time $t$, identifying the velocity amplitude $v_mm$.
15-1 Simple Harmonic Motion (5 of 20) Link to heading
- 15.11 Apply the relationship between velocity amplitude $v_m$, angular frequency $\omega$, and (displacement) $x_m$.
- 15.12 Given an oscillator’s velocity $v(t)$ as a function of time, calculate its acceleration $a(t)$ as a function of time, identify the acceleration amplitude $a_m$ in the result, and calculate the acceleration at any given time.
- 15.13 Sketch a graph of an oscillator’s acceleration $a$ versus time $$, identifying the acceleration amplitude $a_m$.
15-1 Simple Harmonic Motion (6 of 20) Link to heading
- 15.14 Identify that for a simple harmonic oscillator the acceleration $a$ at any instant is always given by the product of a negative constant and the displacement $x$ just then.
- 15.15 For any given instant in an oscillation, apply the relationship between acceleration $a$, angular frequency $\omega$, and displacement $x$.
- 15.16 Given data about the position $x$ and velocity $v$ at one instant determine the phase $\omega t + \phi$ and phase constant $\phi$.
15-1 Simple Harmonic Motion (7 of 20) Link to heading
- 15.17 For a spring-block oscillator, apply the relationships between spring constant k and mass m and either period $T$ or angular frequency $\omega$.
- 15.18 Apply Hooke’s law to relate the force $F$ on a simple harmonic oscillator at any instant to the displacement $x$ of the oscillator at that instant.
15-2 Energy in Simple Harmonic Motion (1 of 3) Link to heading
Learning Objectives
- 15.19 For a spring-block oscillator, calculate the kinetic energy and elastic potential energy at any given time.
- 15.20 Apply the conservation of energy to relate the total energy of a spring-block oscillator at one instant to the total energy at another instant.
- 15.21 Sketch a graph of the kinetic energy, potential energy, and total energy of a spring-block oscillator, first as a function of time and then as a function of the oscillator’s position.
- 15.22 For a spring-block oscillator, determine the block’s position when the total energy is entirely kinetic energy and when it is entirely potential energy.
15-3 An Angular Simple Harmonic Oscillator (1 of 3) Link to heading
Learning Objectives
- 15.23 Describe the motion of an angular simple harmonic oscillator.
- 15.24 For an angular simple harmonic oscillator, apply the relationship between the relationship between $\tau$ and the angular displacement $\theta$ (from equilibrium).
15-3 An Angular Simple Harmonic Oscillator (2 of 3) Link to heading
- 15.25 For an angular simple harmonic oscillator, apply the relationship between the period $T$ (or frequency $f$), the rotational inertia $I$, and the torsion constant
- 15.26 For an angular simple harmonic oscillator at any instant, apply the relationship between the angular acceleration $\alpha$, the angular frequency $\omega$, and the angular displacement $\theta$.
15-4 Pendulums, Circular Motion (1 of 4) Link to heading
Learning Objectives
- 15.27 Describe the motion of an oscillating simple pendulum.
- 15.28 Draw a free-body diagram.
- 15.29-31 Distinguish between a simple and physical pendulum, and relate their variables.
- 15.32 Find angular frequency from torque and angular displacement or acceleration and displacement.
15-4 Pendulums, Circular Motion (2 of 4) Link to heading
- 15.33 Distinguish angular frequency from $\frac{d\theta}{dt}$.
- 15.34 Determine phase and amplitude.
- 15.35 Describe how free-fall acceleration can be measured with a pendulum.
- 15.36 For a physical pendulum, find the center of the oscillation.
- 15.37 Relate SHM to uniform circular motion.
15-5 Damped Simple Harmonic Motion (1 of 7) Link to heading
Learning Objectives
- 15.38 Describe the motion of a damped simple harmonic oscillator and sketch a graph of the oscillator’s position as a function of time.
- 15.39 For any particular time, calculate the position of a damped simple harmonic oscillator.
- 15.40 Determine the amplitude of a damped simple harmonic oscillator at any given time.
15-5 Damped Simple Harmonic Motion (2 of 7) Link to heading
- 15.41 Calculate the angular frequency of a damped simple harmonic oscillator in terms of the spring constant, the damping constant, and the mass, and approximate the angular frequency when the damping constant is small.
- 15.42 Apply the equation giving the (approximate) total energy of a damped simple harmonic oscillator as a function of time.
15-6 Forced Oscillations and Resonance (1 of 7) Link to heading
Learning Objectives
- 15.43 Distinguish between natural angular frequency and driving angular frequency.
- 15.44 For a forced oscillator, sketch a graph of the oscillation amplitude versus the ratio of the driving angular frequency to the natural angular frequency, identify the approximate location of resonance, and indicate the effect of increasing the damping.
- 15.45 For a given natural angular frequency, identify the approximate driving angular frequency that gives resonance.
Summary (1 of 5) Link to heading
- Frequency
- 1 Hz = 1 cyle per second
- Period
- $$\tag{15-2} T = \frac1f $$
- Simple Harmonic Motion
- Find $v$ and $a$ by differentiation $$\tag{15-3} x(t) = x_m \cos(\omega t + \phi) $$ $$\tag{15-5} \omega = \frac{2\pi}{T} = 2\pi f. $$
- The Linear Oscillator.
- $$\tag{15-12} \omega = \sqrt{\frac{k}{m}} $$
- $$\tag{15-13} T = 2\pi \sqrt{\frac{m}{k}} $$
- Energy.
- Pendulums.
- Simple Harmonic Motion and Uniform Circular Motion.
- Damped Harmonic Motion.
- Forced Oscillations and Resonance.
Copyright Link to heading
Copyright © 2018 John Wiley & Sons, Inc.
All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
C. SECTION 117 COMPUTER PROGRAM EXEMPTIONS Link to heading
Section 117 of the Copyright Act of 1976 was enacted in the Computer Software Copyright Amendments of 1980 in response to the recommendations of the National Commission on New Technological Uses of Copyrighted Works’ (CONTU). Section 117 permits the owner of a copy of a computer program to make an additional copy of the program for purely archival purposes if all archival copies are destroyed in the event that continued possession of the computer program should cease to be rightful, or where the making of such a copy is an essential step in the utilization of the computer program in conjunction with a machine and that it is used in no other manner.