Idea: Springs

To review springs again, a spring is defined with some constant $k$ (units $\unit{N/m}$) that results in a force $F=-kx$ when the spring is stretched by a distance $x$. The negative sign indicates that when the spring is stretched some distance, the force the spring exerts tries to bring the spring back to the rest length (or equilibrium length). The potential energy stored in a spring is $U=\frac 12kx^2$.

Definition: Simple Harmonic Motion

Simple Harmonic Motion is a special type of oscillatory motion where the motion follows a sin curve. Whenever we have an equation of motion given by $$a=-\omega^2 x$$ then we say the motion of the object is simple harmonic and oscillates with an angular frequency $\omega$. Note that $\omega$ has units rad/s.

We can imagine simple harmonic motion as the ``shadow" of a bead moving around a vertical circular hoop at an angular speed of $\omega$. This means the frequency of oscillations (number of oscillations per unit time) is $f=\omega/2\pi$ (units 1/s, or Hz), and the period of oscillations (amount of time for one oscillation) is $T=2\pi/\omega$ (units s). For more info, see Giancoli pg 299.

Idea: Spring Oscillations

For a spring, we have $F=-kx$ and $F=ma$, so combining yields $$a=-\frac km x$$ Thus, for a spring, we have the following quantities: $$\omega = \sqrt{\frac km}\qquad f=\frac{1}{2\pi}\sqrt{\frac km}\qquad T=2\pi \sqrt{\frac mk}$$ One side note not mentioned in lecture: these equations all hold true, even if the block is placed in a gravitational field (i.e. vertical spring oscillations).

Idea: Simple Pendulum

A simple pendulum is a bob $m$ hung from a string of length $\ell$. For small horizontal displacements $x$ from the equilibrium position, we can show that the equation of motion follows $$a=-\frac gl x$$so the corresponding SHM values are $$\omega=\sqrt{\frac \ell g}\qquad f=\frac{1}{2\pi}\sqrt{\frac \ell g}\qquad T=2\pi\sqrt{\frac g \ell}$$

Idea: Explicit equation of motion

Since SHM follows a sin curve, we have in general that the dependence of $x$ on $t$ in a situation with SHM is $$x=A\sin(\omega t)$$ where $A$ is the amplitude of oscillations and $\omega$ is the angular frequency.