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_{PH203KUZMASPRING2015}
lenk01.pdf - please download before class!
lenk02.pdf - please download before class!
Please update if an equation is not included
These equations apply only to simple harmonic motion (SHM). The angular frequency $\omega=\frac{2\pi}{T}$.
Starting point: | from equilibrium | from rest^{ (a)} | general^{ (b)}, in terms of sin | general^{ (b)}, in terms of cos |
---|---|---|---|---|
Expression | ||||
Initial phase^{ (}^{c)} | $\theta(0)=-\frac{\pi}{2}$ | $\theta(0)=0$ | $\theta(0)=\phi-\frac{\pi}{2}$ | $\theta(0)=\theta_0$ |
Coordinate, $x$ | $x = x_\text{max}\sin(\omega t)$ | $x = x_\text{max}\cos(\omega t)$ | $x = x_\text{max}\sin(\omega t+\phi)$ | $x = x_\text{max}\cos\!\big(\omega t+\theta_0\!\big)$ |
Velocity, $v$ | $v = x_\text{max}\omega\cos(\omega t)$ | $v = -x_\text{max}\omega\sin(\omega t)$ | $v = x_\text{max}\omega\cos(\omega t+\phi)$ | $v = -x_\text{max}\omega\sin\!\big(\omega t+\theta_0\!\big)$ |
Acceleration, $a$ | $a = -x_\text{max}\omega^2\sin(\omega t)$ | $a = -x_\text{max}\omega^2\cos(\omega t)$ | $a = -x_\text{max}\omega^2\sin(\omega t+\phi)$ | $a = -x_\text{max}\omega^2\cos\!\big(\omega t+\theta_0\!\big)$ |
Kinetic energy, $K$ | $K = \frac{m}{2}x_\text{max}^2\omega^2\cos^2(\omega t)$ | $K = \frac{m}{2}x_\text{max}^2\omega^2\sin^2(\omega t)$ | $K = \frac{m}{2}x_\text{max}^2\omega^2\cos^2(\omega t+\phi)$ | $K = \frac{m}{2}x_\text{max}^2\omega^2\sin^2\!\big(\omega t+\theta_0\!\big)$ |
Pot. energy, $U$ | $U = \frac{m}{2}x_\text{max}^2\omega^2\sin^2(\omega t)$ | $U = \frac{m}{2}x_\text{max}^2\omega^2\cos^2(\omega t)$ | $U = \frac{m}{2}x_\text{max}^2\omega^2\sin^2(\omega t+\phi)$ | $U = \frac{m}{2}x_\text{max}^2\omega^2\cos^2\!\big(\omega t+\theta_0\!\big)$ |
^{a)} Except when the oscillation is caused by an impact. Use “from equilibrium” column in that case ^{b)} The last two columns are equivalent if one assumes $\phi=\theta_0+\frac{\pi}{2}$, because trigonometrically, $\sin\!\big(\theta+\theta_0+\frac{\pi}{2}\!\big)=\cos\!\big(\theta+\theta_0\!\big)$ ^{c)} Initial phase angle measured from the x axis |
Video 1.1: Interesting result from different length pendulums. | Video 1.2: Engineering application of pendulum. 730 ton ball designed to swing inside of an 101 story building. |
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Can we calculate the ratio of individual pendulum lengths from this video? How would we do that? | A nice write up about the TMD… Interesting quote: “The pinnacle [of the building] may oscillate up to 180,000 times a year due to strong wind loads.” This would be a period $T$ of 175 seconds, or a frequency of 0.006 Hz. |
Addition of angles | Doubling of angles | Squaring of trig functions | Small angle $\alpha\ll 1$ |
---|---|---|---|
$\sin(\alpha+\beta)=$ $\sin\alpha\cos\beta+\cos\alpha\sin\beta$ | $\sin(2\alpha)$ $=2\cos\alpha\sin\alpha$ | $\sin^2\alpha=$ $\frac{1}{2}\big(1-\cos(2\alpha)\big)$ | $\sin\alpha$ $\;\approx\; \alpha$ |
$\cos(\alpha+\beta)=$ $\cos\alpha\cos\beta-\sin\alpha\sin\beta$ | $\cos(2\alpha)$ $=\cos^2\alpha-\sin^2\alpha$ $=2\cos^2\alpha-1$ | $\cos^2\alpha=$ $\frac{1}{2}\big(1+\cos(2\alpha)\big)$ | $\cos\alpha\;\approx\;$ $1-\frac{\alpha^2}{2}$ |
Calculate the following:
Steps:
A mouse's heart rate increases from 588 beats per minute (BPM) to 612 BPM in two minutes. What is the change in the heart period?
Steps:
A “pocket balance” (shown to the right) extends by 1/4 of an inch when loaded with a 3 kg mass. Find the frequency with which this mass will oscillate around the equilibrium. Hint: Find the spring constant first.
Steps:
On a similar device, a baby in Mali is being weighed, held by his grandmother. If she were to let go of the infant, he would bounce up and down 3 times per second. Find out how far down does the infant pull the scale when not bouncing. Can you find the baby's weight from these data?
Steps (assuming k is given in the previous problem):
Challenge yourself: can you outline the steps for the same problem if the spring constant k is not known?
It takes 5 lb of force to charge a small spring-loaded toy gun. If fired straight up, it shoots a 10-g projectile 40 feet high. Find the frequency of oscillation if the projectile gets stuck to the spring.
Steps:
Two pendulums (or, pendula) are made of identical 1 kg masses suspended on two weightless strings, 40.0 and 40.5 cm in length. If these pendulums are deflected from vertical by 5 cm and released at the same time, how long will it take for them to get completely “out of step” with each other?
Steps:
A simple pendulum of length $l=1\,$m is deflected from the vertical by $\beta_\text{max}=5^\circ$ and released from rest. Find the velocity when the pendulum passes $\beta=4^\circ$, 3$^\circ$, 2$^\circ$, 1$^\circ$, and 0$^\circ$ positions.
Steps:
Professor, I was reviewing homework problems from chapter 13 and came across one that used the formula, T=2π/square root K/m. But I thought during last lecture you mentioned it could be m/k. Is that true? And if so, in what situations would if differ? — Nichole
A 0.55-kg block slides on a frictionless horizontal surface with a speed of 1.1 m/s . The block encounters an unstretched spring and compresses it 21 cm before coming to rest. (Actual numbers may vary)
What is the force constant?
For what length of time is the block in contact with the spring before it comes to rest?