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# Harmonic Period and Frequency

## Motion of a particle with time that produces a Sinusoidal wave.

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Harmonic Period and Frequency

Students will learn the concepts of period and frequency and how to read them off a graph and how to calculate them for systems in harmonic motion. In addition, students will learn how to graph simple harmonic motion.

### Key Equations

Period EquationsT=1fTspring=2πmkTpendulum=2πLgPeriod is the inverse of frequencyPeriod of mass m on a spring with constant kPeriod of a pendulum of length L\begin{align*} \mathrm{Period~Equations} \begin{cases} T = \frac{1}{f} & \mathrm{Period~is~the~inverse~of~frequency}\\ T_{\mathrm{spring}} = 2\pi\sqrt{\frac{m}{k}} & \mathrm{Period~of~mass~} m \mathrm{~on~a~spring~with~constant~} k\\ T_{\mathrm{pendulum}} = 2\pi\sqrt{\frac{L}{g}} & \mathrm{Period~of~a~pendulum~of~length~} L \end{cases} \end{align*}
Kinematics of SHM{x(t)=x0+Acos2πf(tt0)v(t)=2πfAcos2πf(tt0)Position of an object in SHM of Amplitude AVelocity of an object in SHM of Amplitude A\begin{align*} \mathrm{Kinematics~of~SHM} \begin{cases} x(t) = x_0 + A\cos{2\pi f(t-t_0)} & \mathrm{Position~of~an~object~in~SHM~of~Amplitude~}A\\ v(t) = -2\pi f A\cos{2\pi f(t-t_0)} & \mathrm{Velocity~of~an~object~in~SHM~of~Amplitude~}A \end{cases} \end{align*}

### Guidance

#### Example 1

A bee flaps its wings at a rate of approximately 190 Hz. How long does it take for a bee to flap its wings once (down and up)?

Question: T=?\begin{align*}T = ?\end{align*} [sec]

Given: f=190 Hz\begin{align*}f = 190 \ Hz\end{align*}

Equation: T=1f\begin{align*}T = \frac{1}{f}\end{align*}

Plug n’ Chug: T=1f=1190 Hz=0.00526 s=5.26 ms\begin{align*}T = \frac{1}{f} = \frac{1}{190 \ Hz} = 0.00526 \ s = 5.26 \ ms\end{align*}

#### Example 2

Question: The effective \begin{align*}k\end{align*} of a diving board is \begin{align*}800\mathrm{N/m}\end{align*} (we say effective because it bends in the direction of motion instead of stretching like a spring, but otherwise behaves the same). A pudgy diver is bouncing up and down at the end of the diving board. The y vs. t graph is shown below.

a) What is the distance between the lowest and the highest point of oscillation?

b) What is the Period and frequency of the diver?

c) What is the diver's mass?

d) Write the sinusoidal equation of motion for the diver.

Solution:

a) As we can see from the graph the highest point is 2m and the lowest point is \begin{align*}-2\mathrm{m}\end{align*}. Therefore the distance is \begin{align*} |2\mathrm{m}-(-2\mathrm{m})|=4\mathrm{m} \end{align*}

b)We know that \begin{align*} f=\frac{1}{T} \end{align*} From the graph we know that the period is 2 seconds, so the frequency is \begin{align*}\frac{1}{2 }\end{align*}hz.

c) To find the diver's mass we will use the equation \begin{align*}T=2\pi \sqrt{\frac{m}{k}} \end{align*} and solve for \begin{align*}m\end{align*}. Then it is a simple matter to plug in the known values to get the mass. \begin{align*} T=2\pi \sqrt{\frac{m}{k}} \Rightarrow \frac{T}{2\pi}=\sqrt{\frac{m}{k}} \Rightarrow (\frac{T}{2\pi})^2=\frac{m}{k} \Rightarrow k(\frac{T}{2\pi})^2=m \end{align*} Now we plug in what we know. \begin{align*} m=k(\frac{T}{2\pi})^2=800\frac{N}{m}(\frac{\pi {\mathrm{s}}}{2\pi})^2=200\mathrm{kg} \end{align*}

d) To get the sinusoidal equation we must first choose whether to go with a cosine graph or a sine graph. Then we must find the amplitude (A), vertical shift (D), horizontal shift (C), and period (B). Cosine is easier in this case so we will work with it instead of sine. As we can see from the graph, the amplitude is 2, the vertical shift is 0, and the horizontal shift is \begin{align*}-0.4\end{align*}. We solved for the period already. Therefore, we can write the sinusoidal equation of this graph. \begin{align*} AcosB(x-C)+D=2cos\pi(x+.4) \end{align*}

### Explore More

1. While treading water, you notice a buoy way out towards the horizon. The buoy is bobbing up and down in simple harmonic motion. You only see the buoy at the most upward part of its cycle. You see the buoy appear 10 times over the course of one minute.
1. What kind of force that is leading to simple harmonic motion?
2. What is the period \begin{align*}(T)\end{align*} and frequency \begin{align*}(f)\end{align*} of its cycle? Use the proper units.
2. The pitch of a Middle \begin{align*}C\end{align*} note on a piano is \begin{align*}263\;\mathrm{Hz}\end{align*}. This means when you hear this note, the hairs in your ears wiggle back and forth at this frequency.
1. What is the period of oscillation for your ear hairs?
2. What is the period of oscillation of the struck wire within the piano?
3. The Sun tends to have dark, Earth-sized spots on its surface due to kinks in its magnetic field. The number of visible spots varies over the course of years. Use the graph of the sunspot cycle below to answer the following questions. (Note that this is real data from our sun, so it doesn’t look like a perfect sine wave. What you need to do is estimate the best sine wave that fits this data.)
1. Estimate the period \begin{align*}T\end{align*} in years.
2. When do we expect the next “solar maximum?”

#### Answers to Selected Problems

1. a. Buoyant force and gravity b. \begin{align*}T = 6 \;\mathrm{s}, f = 1/6 \;\mathrm{Hz}\end{align*}
2. a. \begin{align*}0.0038 \;\mathrm{s}\end{align*} b. \begin{align*}0.0038 \;\mathrm{s}\end{align*}
3. a. About 11 years b. About 2014