Notes
The SHM differential equation
Simple harmonic motion happens when $\displaystyle \frac{d^2x}{dt^2} = -\omega^2x$. The solution is
with angular frequency $\omega$, period $\displaystyle T = \frac{2\pi}{\omega}$, and frequency $\displaystyle f = \frac{\omega}{2\pi}$.
- Velocity $v = -A\omega\sin(\omega t + \phi)$; maximum $v_{max} = A\omega$.
- Acceleration $a = -A\omega^2\cos(\omega t + \phi)$; maximum $a_{max} = A\omega^2$.
Systems
| System | $\omega$ | Period |
|---|---|---|
| Mass on a spring | $\displaystyle \sqrt{\frac{k}{m}}$ | $\displaystyle 2\pi\sqrt{\frac{m}{k}}$ |
| Simple pendulum (small angle) | $\displaystyle \sqrt{\frac{g}{\ell}}$ | $\displaystyle 2\pi\sqrt{\frac{\ell}{g}}$ |
| Physical pendulum | $\displaystyle \sqrt{\frac{mgd}{I}}$ | $\displaystyle 2\pi\sqrt{\frac{I}{mgd}}$ |
- For the physical pendulum, $d$ is the distance from the pivot to the center of mass.
- A vertical spring oscillates with the same period as a horizontal one; it just has a new equilibrium position.
- Springs in series: $\displaystyle \frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2}$. In parallel: $k_{eq} = k_1 + k_2$.
Energy
$\displaystyle E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$. Speed at any position: $\displaystyle v = \omega\sqrt{A^2 - x^2}$.
Practice
$\displaystyle T = \frac{2\pi}{5} \approx 1.26$ s.
$v_{max} = A\omega = 0.20(5)$.
Parallel springs add.
$\displaystyle I = \frac{1}{3}ML^2$ and $\displaystyle d = \frac{L}{2}$: $\displaystyle \frac{I}{Mgd} = \frac{2L}{3g}$.
Gravity only shifts the equilibrium position.
The acceleration is proportional to displacement and opposite it.
$\displaystyle v = 10\sqrt{0.010 - 0.0036} = 10(0.080)$.
$a_{max} = A\omega^2 = 0.05(400)$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A block of mass $m$ on a frictionless surface is attached to two springs, one on each side, with constants $k_1$ and $k_2$. Both springs are at their natural lengths when the block is at $x = 0$.
(a) Write Newton's second law for the block when it is displaced by $x$.
Show the answer and scoring guide
Both springs push the block back: $\displaystyle m\frac{d^2x}{dt^2} = -k_1x - k_2x = -(k_1 + k_2)x$.
Check off the points you earned:
(b) Show that the motion is SHM and find the period.
Show the answer and scoring guide
$\displaystyle \frac{d^2x}{dt^2} = -\frac{k_1 + k_2}{m}x$, which has the SHM form with $\displaystyle \omega^2 = \frac{k_1 + k_2}{m}$. $\displaystyle T = 2\pi\sqrt{\frac{m}{k_1 + k_2}}$.
Check off the points you earned:
(c) The block is released from rest at $x = A$. Write $x(t)$.
Show the answer and scoring guide
$\displaystyle x(t) = A\cos\left(\sqrt{\frac{k_1 + k_2}{m}}\,t\right)$.
Check off the points you earned: