# 15.S: Oscillations (Summary)

### Key Terms

amplitude (A) | maximum displacement from the equilibrium position of an object oscillating around the equilibrium position |

critically damped | 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 |

elastic potential energy | potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring |

equilibrium position | position where the spring is neither stretched nor compressed |

force constant (k) | characteristic of a spring which is defined as the ratio of the force applied to the spring to the displacement caused by the force |

frequency (f) | number of events per unit of time |

natural angular frequency | angular frequency of a system oscillating in SHM |

oscillation | single fluctuation of a quantity, or repeated and regular fluctuations of a quantity, between two extreme values around an equilibrium or average value |

overdamped | 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 |

period (T) | time taken to complete one oscillation |

periodic motion | motion that repeats itself at regular time intervals |

phase shift | 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 |

physical pendulum | any extended object that swings like a pendulum |

resonance | large amplitude oscillations in a system produced by a small amplitude driving force, which has a frequency equal to the natural frequency |

restoring force | force acting in opposition to the force caused by a deformation |

simple harmonic motion (SHM) | oscillatory motion in a system where the restoring force is proportional to the displacement, which acts in the direction opposite to the displacement |

simple harmonic oscillator | a device that oscillates in SHM where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement |

simple pendulum | point mass, called a pendulum bob, attached to a near massless string |

stable equilibrium point | 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 |

torsional pendulum | any suspended object that oscillates by twisting its suspension |

underdamped | condition in which damping of an oscillator causes the amplitude of oscillations of a damped harmonic oscillator to decrease over time, eventually approaching zero |

### Key Equations

Relationship between frequency and period | $$f = \frac{1}{T}$$ |

Position in SHM with \(\phi\) = 0.00 | $$x(t) = A \cos (\omega t)$$ |

General position in SHM | $$x(t) = A \cos (\omega t + \phi)$$ |

General velocity in SHM | $$v(t) = -A \omega \sin (\omega t + \phi)$$ |

General acceleration in SHM | $$a(t) = -A \omega^{2} \cos (\omega t + \phi)$$ |

Maximum displacement (amplitude) of SHM | $$x_{max} = A$$ |

Maximum velocity of SHM | $$|v_{max}| = A \omega$$ |

Maximum acceleration of SHM | $$|a_{max}| = A \omega^{2}$$ |

Angular frequency of a mass-spring system in SHM | $$\omega = \sqrt{\frac{k}{m}}$$ |

Period of a mass-spring system in SHM | $$T = 2 \pi \sqrt{\frac{m}{k}}$$ |

Frequency of a mass-spring system in SHM | $$f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}$$ |

Energy in a mass-spring system in SHM | $$E_{Total} = \frac{1}{2} kx^{2} + \frac{1}{2} mv^{2} = \frac{1}{2} kA^{2}$$ |

The velocity of the mass in a spring-mass system in SHM | $$v = \pm \sqrt{\frac{k}{m} (A^{2} - x^{2})}$$ |

The x-component of the radius of a rotating disk | $$x(t) = A \cos (\omega t + \phi)$$ |

The x-component of the velocity of the edge of a rotating disk | $$v(t) = -v_{max} \sin (\omega t + \phi)$$ |

The x-component of the acceleration of the edge of a rotating disk | $$a(t) = -a_{max} \cos (\omega t + \phi)$$ |

Force equation for a simple pendulum | $$\frac{d^{2} \theta}{dt^{2}} = - \frac{g}{L} \theta$$ |

Angular frequency for a simple pendulum | $$\omega = \sqrt{\frac{g}{L}}$$ |

Period of a simple pendulum | $$T = 2 \pi \sqrt{\frac{L}{g}}$$ |

Angular frequency of a physical pendulum | $$\omega = \sqrt{\frac{mgL}{I}}$$ |

Period of a physical pendulum | $$T = 2 \pi \sqrt{\frac{I}{mgL}}$$ |

Period of a torsional pendulum | $$T = 2 \pi \sqrt{\frac{I}{\kappa}}$$ |

Newton’s second law for harmonic motion | $$m \frac{d^{2} x}{dt^{2}} + b \frac{dx}{dt} + kx = 0$$ |

Solution for underdamped harmonic motion | $$x(t) = A_{0} e^{- \frac{b}{2m} t} \cos (\omega t + \phi)$$ |

Natural angular frequency of a mass-spring system | $$\omega_{0} = \sqrt{\frac{k}{m}}$$ |

Angular frequency of underdamped harmonic motion | $$\omega = \sqrt{\omega_{0}^{2} - \left(\dfrac{b}{2m}\right)^{2}}$$ |

Newton’s second law for forced, damped oscillation | $$-kx -b \frac{dx}{dt} + F_{0} \sin (\omega t) = m \frac{d^{2} x}{dt^{2}}$$ |

Solution to Newton’s second law for forced, damped oscillations | $$x(t) = A \cos (\omega t + \phi)$$ |

Amplitude of system undergoing forced, damped oscillations | $$A = \frac{F_{0}}{\sqrt{m (\omega^{2} - \omega_{0}^{2})^{2} + b^{2} \omega^{2}}}$$ |

### Summary

#### 15.1 Simple Harmonic Motion

- Periodic motion is a repeating oscillation. The time for one oscillation is the period T and the number of oscillations per unit time is the frequency f. These quantities are related by \(f = \frac{1}{T}\).
- Simple harmonic motion (SHM) is oscillatory motion for a system where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement.
- Maximum displacement is the amplitude A. The angular frequency \(\omega\), period T, and frequency f of a simple harmonic oscillator are given by \(\omega = \sqrt{\frac{k}{m}}\), T = 2\(\pi \sqrt{\frac{m}{k}}\), and f = \(\frac{1}{2 \pi} \sqrt{\frac{k}{m}}\), where m is the mass of the system and k is the force constant.
- Displacement as a function of time in SHM is given by x(t) = Acos\(\left(\dfrac{2 \pi}{T} t + \phi \right)\) = Acos(\(\omega t + \phi\)).
- The velocity is given by v(t) = -A\(\omega\)sin(\(\omega t + \phi\)) = -v
_{max}sin(\(\omega t + \phi\)), where v_{max}= A\(\omega\) = A\(\sqrt{\frac{k}{m}}\). - The acceleration is given by a(t) = -A\(\omega^{2}\)cos(\(\omega t + \phi\)) = -a
_{max}cos(\(\omega t + \phi\)), where a_{max}= A\(\omega^{2}\) = A\(\frac{k}{m}\).

#### 15.2 Energy in Simple Harmonic Motion

- The simplest type of oscillations are related to systems that can be described by Hooke’s law, F = −kx, where F is the restoring force, x is the displacement from equilibrium or deformation, and k is the force constant of the system.
- Elastic potential energy U stored in the deformation of a system that can be described by Hooke’s law is given by U = \(\frac{1}{2}\)kx
^{2}. - Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant: $$E_{Total} = \frac{1}{2} kx^{2} + \frac{1}{2} mv^{2} = \frac{1}{2} kA^{2} = constant \ldotp$$
- The magnitude of the velocity as a function of position for the simple harmonic oscillator can be found by using $$v = \sqrt{\frac{k}{m} (A^{2} - x^{2})} \ldotp$$

#### 15.3 Comparing Simple Harmonic Motion and Circular Motion

- A projection of uniform circular motion undergoes simple harmonic oscillation.
- Consider a circle with a radius A, moving at a constant angular speed \(\omega\). A point on the edge of the circle moves at a constant tangential speed of v
_{max}= A\(\omega\). The projection of the radius onto the x-axis is x(t) = Acos(\(\omega\)t + \(\phi\)), where (\(\phi\)) is the phase shift. The x-component of the tangential velocity is v(t) = −A\(\omega\)sin(\(\omega\)t + \(\phi\)).

#### 15.4 Pendulums

- A mass m suspended by a wire of length L and negligible mass is a simple pendulum and undergoes SHM for amplitudes less than about 15°. The period of a simple pendulum is T = 2\(\pi \sqrt{\frac{L}{g}}\), where L is the length of the string and g is the acceleration due to gravity.
- The period of a physical pendulum T = 2\(\pi \sqrt{\frac{I}{mgL}}\) can be found if the moment of inertia is known. The length between the point of rotation and the center of mass is L.
- The period of a torsional pendulum T = 2\(\pi \sqrt{\frac{I}{\kappa}}\) can be found if the moment of inertia and torsion constant are known.

#### 15.5 Damped Oscillations

- Damped harmonic oscillators have non-conservative forces that dissipate their energy.
- Critical damping returns the system to equilibrium as fast as possible without overshooting.
- An underdamped system will oscillate through the equilibrium position.
- An overdamped system moves more slowly toward equilibrium than one that is critically damped.

#### 15.6 Forced Oscillations

- A system’s natural frequency is the frequency at which the system oscillates if not affected by driving or damping forces.
- A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate.
- The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.

### Contributors

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