{"id":1499,"date":"2018-02-06T17:56:03","date_gmt":"2018-02-06T17:56:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/?post_type=chapter&p=1499"},"modified":"2018-02-26T17:04:08","modified_gmt":"2018-02-26T17:04:08","slug":"15-chapter-review","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/15-chapter-review\/","title":{"raw":"15 Chapter Review","rendered":"15 Chapter Review"},"content":{"raw":"
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Key Terms<\/span><\/h3>\r\n
\r\n \t
amplitude (A<\/em>)<\/strong><\/dt>\r\n \t
maximum displacement from the equilibrium position of an object oscillating around the equilibrium position<\/dd>\r\n<\/dl>\r\n
\r\n \t
critically damped<\/strong><\/dt>\r\n \t
condition in which the damping of an oscillator causes it to return as quickly as possible to its equilibrium position without oscillating back and forth about this position<\/dd>\r\n<\/dl>\r\n
\r\n \t
elastic potential energy<\/strong><\/dt>\r\n \t
potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring<\/dd>\r\n<\/dl>\r\n
\r\n \t
equilibrium position<\/strong><\/dt>\r\n \t
position where the spring is neither stretched nor compressed<\/dd>\r\n<\/dl>\r\n
\r\n \t
force constant (k<\/em>)<\/strong><\/dt>\r\n \t
characteristic of a spring which is defined as the ratio of the force applied to the spring to the displacement caused by the force<\/dd>\r\n<\/dl>\r\n
\r\n \t
frequency (f<\/em>)<\/strong><\/dt>\r\n \t
number of events per unit of time<\/dd>\r\n<\/dl>\r\n
\r\n \t
natural angular frequency<\/strong><\/dt>\r\n \t
angular frequency of a system oscillating in SHM<\/dd>\r\n<\/dl>\r\n
\r\n \t
oscillation<\/strong><\/dt>\r\n \t
single fluctuation of a quantity, or repeated and regular fluctuations of a quantity, between two extreme values around an equilibrium or average value<\/dd>\r\n<\/dl>\r\n
\r\n \t
overdamped<\/strong><\/dt>\r\n \t
condition in which damping of an oscillator causes it to return to equilibrium without oscillating; oscillator moves more slowly toward equilibrium than in the critically damped system<\/dd>\r\n<\/dl>\r\n
\r\n \t
period (T<\/em>)<\/strong><\/dt>\r\n \t
time taken to complete one oscillation<\/dd>\r\n<\/dl>\r\n
\r\n \t
periodic motion<\/strong><\/dt>\r\n \t
motion that repeats itself at regular time intervals<\/dd>\r\n<\/dl>\r\n
\r\n \t
phase shift<\/strong><\/dt>\r\n \t
angle, in radians, that is used in a cosine or sine function to shift the function left or right, used to match up the function with the initial conditions of data<\/dd>\r\n<\/dl>\r\n
\r\n \t
physical pendulum<\/strong><\/dt>\r\n \t
any extended object that swings like a pendulum<\/dd>\r\n<\/dl>\r\n
\r\n \t
resonance<\/strong><\/dt>\r\n \t
large amplitude oscillations in a system produced by a small amplitude driving force, which has a frequency equal to the natural frequency<\/dd>\r\n<\/dl>\r\n
\r\n \t
restoring force<\/strong><\/dt>\r\n \t
force acting in opposition to the force caused by a deformation<\/dd>\r\n<\/dl>\r\n
\r\n \t
simple harmonic motion (SHM)<\/strong><\/dt>\r\n \t
oscillatory motion in a system where the restoring force is proportional to the displacement, which acts in the direction opposite to the displacement<\/dd>\r\n<\/dl>\r\n
\r\n \t
simple harmonic oscillator<\/strong><\/dt>\r\n \t
a device that oscillates in SHM where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement<\/dd>\r\n<\/dl>\r\n
\r\n \t
simple pendulum<\/strong><\/dt>\r\n \t
point mass, called a pendulum bob, attached to a near massless string<\/dd>\r\n<\/dl>\r\n
\r\n \t
stable equilibrium point<\/strong><\/dt>\r\n \t
point where the net force on a system is zero, but a small displacement of the mass will cause a restoring force that points toward the equilibrium point<\/dd>\r\n<\/dl>\r\n
\r\n \t
torsional pendulum<\/strong><\/dt>\r\n \t
any suspended object that oscillates by twisting its suspension<\/dd>\r\n<\/dl>\r\n
\r\n \t
underdamped<\/strong><\/dt>\r\n \t
condition in which damping of an oscillator causes the amplitude of oscillations of a damped harmonic oscillator to decrease over time, eventually approaching zero<\/dd>\r\n<\/dl>\r\n<\/div>\r\n \r\n\r\n<\/div>\r\n
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Key Equations<\/span><\/h3>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Relationship between frequency and period<\/td>\r\nf<\/span>=<\/span>1<\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span>f=1T<\/span><\/span><\/td>\r\n<\/tr>\r\n
Position in SHM with<\/span><\/span>\u03d5<\/span>=<\/span>0.00<\/span><\/span><\/span><\/span><\/span><\/span>Position in SHM with\u03d5=0.00<\/span><\/span><\/td>\r\nx<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span>A<\/span><\/span>cos<\/span>(<\/span>\u03c9<\/span>t<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>x(t)=Acos(\u03c9t)<\/span><\/span><\/td>\r\n<\/tr>\r\n
General position in SHM<\/td>\r\nx<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span>A<\/span>cos<\/span>(<\/span>\u03c9<\/span>t<\/span>+<\/span>\u03d5<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>x(t)=Acos(\u03c9t+\u03d5)<\/span><\/span><\/td>\r\n<\/tr>\r\n
General velocity in SHM<\/td>\r\nv<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span>\u2212<\/span>A<\/span>\u03c9<\/span>sin<\/span>(<\/span>\u03c9<\/span>t<\/span>+<\/span>\u03d5<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>v(t)=\u2212A\u03c9sin(\u03c9t+\u03d5)<\/span><\/span><\/td>\r\n<\/tr>\r\n
General acceleration in SHM<\/td>\r\na<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span>\u2212<\/span>A<\/span>\u03c9<\/span>2<\/span><\/span>cos<\/span>(<\/span>\u03c9<\/span>t<\/span>+<\/span>\u03d5<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>a(t)=\u2212A\u03c92cos(\u03c9t+\u03d5)<\/span><\/span><\/td>\r\n<\/tr>\r\n
Maximum displacement (amplitude) of SHM<\/td>\r\nx<\/span>max<\/span><\/span><\/span>=<\/span>A<\/span><\/span><\/span><\/span><\/span><\/span>xmax=A<\/span><\/span><\/td>\r\n<\/tr>\r\n
Maximum velocity of SHM<\/td>\r\n\u2223\u2223<\/span>v<\/span>max<\/span><\/span><\/span><\/span>\u2223\u2223<\/span><\/span>=<\/span>A<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span>|vmax|=A\u03c9<\/span><\/span><\/td>\r\n<\/tr>\r\n
Maximum acceleration of SHM<\/td>\r\n\u2223\u2223<\/span>a<\/span>max<\/span><\/span><\/span><\/span>\u2223\u2223<\/span><\/span>=<\/span>A<\/span>\u03c9<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>|amax|=A\u03c92<\/span><\/span><\/td>\r\n<\/tr>\r\n
Angular frequency of a mass-spring system in SHM<\/td>\r\n\u03c9<\/span>=<\/span>k<\/span>m<\/span><\/span><\/span><\/span>\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>\u03c9=km<\/span><\/span><\/td>\r\n<\/tr>\r\n
Period of a mass-spring system in SHM<\/td>\r\nT<\/span>=<\/span>2<\/span>\u03c0<\/span>m<\/span>k<\/span><\/span><\/span><\/span>\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>T=2\u03c0mk<\/span><\/span><\/td>\r\n<\/tr>\r\n
Frequency of a mass-spring system in SHM<\/td>\r\nf<\/span>=<\/span>1<\/span>2<\/span>\u03c0<\/span><\/span><\/span>k<\/span>m<\/span><\/span><\/span><\/span>\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>f=12\u03c0km<\/span><\/span><\/td>\r\n<\/tr>\r\n
Energy in a mass-spring system in SHM<\/td>\r\nE<\/span>Total<\/span><\/span><\/span>=<\/span>1<\/span>2<\/span><\/span>k<\/span>x<\/span>2<\/span><\/span>+<\/span>1<\/span>2<\/span><\/span>m<\/span>v<\/span>2<\/span><\/span>=<\/span>1<\/span>2<\/span><\/span>k<\/span>A<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>ETotal=12kx2+12mv2=12kA2<\/span><\/span><\/td>\r\n<\/tr>\r\n
The velocity of the mass in a spring-mass\r\n
<\/div>\r\nsystem in SHM<\/td>\r\n
v<\/span>=<\/span>\u00b1<\/span>k<\/span>m<\/span><\/span>(<\/span>A<\/span>2<\/span><\/span>\u2212<\/span>x<\/span>2<\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>v=\u00b1km(A2\u2212x2)<\/span><\/span><\/td>\r\n<\/tr>\r\n
The\u00a0x<\/em>-component of the radius of a rotating disk<\/td>\r\nx<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span>A<\/span>cos<\/span>(<\/span>\u03c9<\/span><\/span>t<\/span>+<\/span>\u03d5<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>x(t)=Acos(\u03c9t+\u03d5)<\/span><\/span><\/td>\r\n<\/tr>\r\n
The\u00a0x<\/em>-component of the velocity of the edge of a rotating disk<\/td>\r\nv<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span>\u2212<\/span>v<\/span>max<\/span><\/span><\/span>sin<\/span>(<\/span>\u03c9<\/span><\/span>t<\/span>+<\/span>\u03d5<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>v(t)=\u2212vmaxsin(\u03c9t+\u03d5)<\/span><\/span><\/td>\r\n<\/tr>\r\n
The\u00a0x<\/em>-component of the acceleration of the\r\n
<\/div>\r\nedge of a rotating disk<\/td>\r\n
a<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span>\u2212<\/span>a<\/span>max<\/span><\/span><\/span>cos<\/span>(<\/span>\u03c9<\/span><\/span>t<\/span>+<\/span>\u03d5<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>a(t)=\u2212amaxcos(\u03c9t+\u03d5)<\/span><\/span><\/td>\r\n<\/tr>\r\n
Force equation for a simple pendulum<\/td>\r\nd<\/span>2<\/span><\/span>\u03b8<\/span><\/span>d<\/span>t<\/span>2<\/span><\/span><\/span><\/span>=<\/span>\u2212<\/span>g<\/span>L<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span>d2\u03b8dt2=\u2212gL\u03b8<\/span><\/span><\/td>\r\n<\/tr>\r\n
Angular frequency for a simple pendulum<\/td>\r\n\u03c9<\/span>=<\/span>g<\/span>L<\/span><\/span><\/span><\/span>\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>\u03c9=gL<\/span><\/span><\/td>\r\n<\/tr>\r\n
Period of a simple pendulum<\/td>\r\nT<\/span>=<\/span>2<\/span>\u03c0<\/span>L<\/span>g<\/span><\/span><\/span><\/span>\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>T=2\u03c0Lg<\/span><\/span><\/td>\r\n<\/tr>\r\n
Angular frequency of a physical pendulum<\/td>\r\n\u03c9<\/span>=<\/span>m<\/span>g<\/span>L<\/span><\/span>I<\/span><\/span><\/span><\/span>\u203e\u203e\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>\u03c9=mgLI<\/span><\/span><\/td>\r\n<\/tr>\r\n
Period of a physical pendulum<\/td>\r\nT<\/span>=<\/span>2<\/span>\u03c0<\/span>I<\/span>m<\/span>g<\/span>L<\/span><\/span><\/span><\/span><\/span>\u203e\u203e\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>T=2\u03c0ImgL<\/span><\/span><\/td>\r\n<\/tr>\r\n
Period of a torsional pendulum<\/td>\r\nT<\/span>=<\/span>2<\/span>\u03c0<\/span>I<\/span>\u03ba<\/span><\/span><\/span><\/span>\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>T=2\u03c0I\u03ba<\/span><\/span><\/td>\r\n<\/tr>\r\n
Newton\u2019s second law for harmonic motion<\/td>\r\nm<\/span>d<\/span>2<\/span><\/span>x<\/span><\/span>d<\/span>t<\/span>2<\/span><\/span><\/span><\/span>+<\/span>b<\/span>d<\/span>x<\/span><\/span>d<\/span>t<\/span><\/span><\/span>+<\/span>k<\/span>x<\/span>=<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span>md2xdt2+bdxdt+kx=0<\/span><\/span><\/td>\r\n<\/tr>\r\n
Solution for underdamped harmonic motion<\/td>\r\nx<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span>A<\/span>0<\/span><\/span>e<\/span>\u2212<\/span>b<\/span>2<\/span>m<\/span><\/span><\/span>t<\/span><\/span><\/span>cos<\/span>(<\/span>\u03c9<\/span>t<\/span>+<\/span>\u03d5<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>x(t)=A0e\u2212b2mtcos(\u03c9t+\u03d5)<\/span><\/span><\/td>\r\n<\/tr>\r\n
Natural angular frequency of a\r\n
<\/div>\r\nmass-spring system<\/td>\r\n
\u03c9<\/span>0<\/span><\/span>=<\/span>k<\/span>m<\/span><\/span><\/span><\/span>\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>\u03c90=km<\/span><\/span><\/td>\r\n<\/tr>\r\n
Angular frequency of underdamped\r\n
<\/div>\r\nharmonic motion<\/td>\r\n
\u03c9<\/span>=<\/span>\u03c9<\/span>2<\/span>0<\/span><\/span>\u2212<\/span>(<\/span>b<\/span>2<\/span>m<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>\u03c9=\u03c902\u2212(b2m)2<\/span><\/span><\/td>\r\n<\/tr>\r\n
Newton\u2019s second law for forced,\r\n
<\/div>\r\ndamped oscillation<\/td>\r\n
\u2212<\/span>k<\/span>x<\/span>\u2212<\/span>b<\/span>d<\/span>x<\/span><\/span>d<\/span>t<\/span><\/span><\/span>+<\/span>F<\/span>o<\/span><\/span>sin<\/span>(<\/span>\u03c9<\/span>t<\/span><\/span>)<\/span><\/span>=<\/span>m<\/span>d<\/span>2<\/span><\/span>x<\/span><\/span>d<\/span>t<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212kx\u2212bdxdt+Fosin(\u03c9t)=md2xdt2<\/span><\/span><\/td>\r\n<\/tr>\r\n
Solution to Newton\u2019s second law for forced,\r\n
<\/div>\r\ndamped oscillations<\/td>\r\n
x<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span>A<\/span>cos<\/span>(<\/span>\u03c9<\/span>t<\/span>+<\/span>\u03d5<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>x(t)=Acos(\u03c9t+\u03d5)<\/span><\/span><\/td>\r\n<\/tr>\r\n
Amplitude of system undergoing forced,\r\n
<\/div>\r\ndamped oscillations<\/td>\r\n
A<\/span>=<\/span>F<\/span>o<\/span><\/span><\/span>m<\/span>(<\/span>\u03c9<\/span>2<\/span><\/span>\u2212<\/span>\u03c9<\/span>2<\/span>o<\/span><\/span><\/span>)<\/span><\/span><\/span>2<\/span><\/span>+<\/span>b<\/span>2<\/span><\/span>\u03c9<\/span>2<\/span><\/span><\/span><\/span>\u221a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A=Fom(\u03c92\u2212\u03c9o2)2+b2\u03c92<\/span><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><\/div>\r\n \r\n\r\n<\/div>\r\n
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Summary<\/span><\/h3>\r\n
\r\n
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15.1<\/span>\u00a0<\/span>Simple Harmonic Motion<\/span><\/h4>\r\n