### 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 $\omega =\sqrt{\frac{k}{m}}$, $T=2\pi \sqrt{\frac{m}{k}},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}f=\frac{1}{2\pi}\sqrt{\frac{k}{m}}$, 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$x(t)=A\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\frac{2\pi}{T}t+\varphi \right)=A\text{cos}\left(\omega t+\varphi \right)$.
- The velocity is given by $v(t)=\text{\u2212}A\omega \text{sin}\left(\omega t+\varphi \right)=\text{\u2212}{v}_{\text{max}}\text{sin}\left(\omega t+\varphi \right),\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}{\text{v}}_{\text{max}}=A\omega =A\sqrt{\frac{k}{m}}$.
- The acceleration is $a(t)=\text{\u2212}A{\omega}^{2}\text{cos}\left(\omega t+\varphi \right)=\text{\u2212}{a}_{\text{max}}\text{cos}\left(\omega t+\varphi \right)$, where ${a}_{\text{max}}=A{\omega}^{2}=A\frac{k}{m}$.

### 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$U=\frac{1}{2}k{x}^{2}.$ - Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
$${E}_{\text{Total}}=\frac{1}{2}m{v}^{2}+\frac{1}{2}k{x}^{2}=\frac{1}{2}k{A}^{2}=\text{constant.}$$
- The magnitude of the velocity as a function of position for the simple harmonic oscillator can be found by using
$$\left|v\right|=\sqrt{\frac{k}{m}\left({A}^{2}-{x}^{2}\right)}.$$

### 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}_{\text{max}}=A\omega $. The projection of the radius onto the*x*-axis is $x\left(t\right)=A\text{cos}\left(\omega t+\varphi \right)$, where $\left(\varphi \right)$ is the phase shift. The*x*-component of the tangential velocity is $v\left(t\right)=\text{\u2212}A\omega \text{sin}\left(\omega t+\varphi \right)$.

### 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 $15\text{\xb0}$. 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.