#set document(title: "16.2 Period and Frequency in Oscillations", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 16.2#h(0.6em)Period and Frequency in Oscillations === Learning Objectives By the end of this section, you will be able to: - Observe the vibrations of a guitar string. - Determine the frequency of oscillations. #figure(figph[The given figure shows a closed zoom view of the strings of a guitar. There are two slanting white colored strings in the picture. In the nearer string, the gaps between the circular threads of the string are visible, whereas the second white string at the back looks like a white thin stick.], alt: "The given figure shows a closed zoom view of the strings of a guitar. There are two slanting white colored strings in the picture. In the nearer string, the gaps between the circular threads of the string are visible, whereas the second white string at the back looks like a white thin stick.", caption: [The strings on this guitar vibrate at regular time intervals.]) When you pluck a guitar string, the resulting sound has a steady tone and lasts a long time. Each successive vibration of the string takes the same time as the previous one. We define #strong[periodic motion] to be a motion that repeats itself at regular time intervals, such as exhibited by the guitar string or by an object on a spring moving up and down. The time to complete one oscillation remains constant and is called the #strong[period] #math.equation(block: false, alt: "T")[$T$]. Its units are usually seconds, but may be any convenient unit of time. The word period refers to the time for some event whether repetitive or not; but we shall be primarily interested in periodic motion, which is by definition repetitive. A concept closely related to period is the frequency of an event. For example, if you get a paycheck twice a month, the frequency of payment is two per month and the period between checks is half a month. #strong[Frequency] #math.equation(block: false, alt: "f")[$f$] is defined to be the number of events per unit time. For periodic motion, frequency is the number of oscillations per unit time. The relationship between frequency and period is #math.equation(block: true, alt: "f equals the fraction 1 over T .")[$f = frac(1, T) .$] The SI unit for frequency is the #emph[cycle per second], which is defined to be a #emph[hertz] (Hz): #math.equation(block: true, alt: "1 Hz equals 1 the fraction cycle over s or 1 Hz equals the fraction 1 over s")[$" 1 Hz " = 1 frac(" cycle ", " s ") " or 1 Hz " = frac(1, " s ")$] A cycle is one complete oscillation. Note that a vibration can be a single or multiple event, whereas oscillations are usually repetitive for a significant number of cycles. #examplebox("Example 1")[Determine the Frequency of Two Oscillations: Medical Ultrasound and the Period of Middle C][ We can use the formulas presented in this module to determine both the frequency based on known oscillations and the oscillation based on a known frequency. Let’s try one example of each. (a) A medical imaging device produces ultrasound by oscillating with a period of 0.400 µs. What is the frequency of this oscillation? (b) The frequency of middle C on a typical musical instrument is 264 Hz. What is the time for one complete oscillation? Strategy Both questions (a) and (b) can be answered using the relationship between period and frequency. In question (a), the period #emph[#math.equation(block: false, alt: "T")[$T$]] is given and we are asked to find frequency #math.equation(block: false, alt: "f")[$f$]. In question (b), the frequency #math.equation(block: false, alt: "f")[$f$] is given and we are asked to find the period #emph[#math.equation(block: false, alt: "T")[$T$]]. Solution a + Substitute #math.equation(block: false, alt: "0 . 400 μ s")[$0 " . " " 400 " #h(0.25em) " μ " " s"$] for #emph[#math.equation(block: false, alt: "T")[$T$]] in #math.equation(block: false, alt: "f equals the fraction 1 over T")[$f = frac(1, T)$]: #math.equation(block: true, alt: "f equals the fraction 1 over T equals the fraction 1 over 0 . 400 times 10 to the power minus 6 s .")[$f = frac(1, T) = frac(1, 0 "." "400" × "10"^(− 6) #h(0.25em) s) .$]Solve to find #math.equation(block: true, alt: "f equals 2 . 50 times 10 to the power 6 Hz .")[$f = 2 "." "50" × attach(" 10", t: 6) #h(0.25em) "Hz" .$] Discussion a The frequency of sound found in (a) is much higher than the highest frequency that humans can hear and, therefore, is called ultrasound. Appropriate oscillations at this frequency generate ultrasound used for noninvasive medical diagnoses, such as observations of a fetus in the womb. Solution b + Identify the known values: The time for one complete oscillation is the period #emph[#math.equation(block: false, alt: "T")[$T$]]: #math.equation(block: true, alt: "f equals the fraction 1 over T .")[$f = frac(1, T) .$] + Solve for #emph[#math.equation(block: false, alt: "T")[$T$]]: #math.equation(block: true, alt: "T equals the fraction 1 over f .")[$T = frac(1, f) .$] + Substitute the given value for the frequency into the resulting expression: #math.equation(block: true, alt: "T equals the fraction 1 over f equals the fraction 1 over 264 Hz equals the fraction 1 over 264 cycles/s equals 3 . 79 times 10 to the power minus 3 s equals 3 . 79 ms .")[$T = frac(1, f) = frac(1, "264" #h(0.25em) " Hz") = frac(1, "264" #h(0.25em) " cycles/s") = 3 "." "79" × "10"^(− 3) #h(0.25em) "s" = 3 "." "79" #h(0.25em) " ms" .$] Discussion b The period found in (b) is the time per cycle, but this value is often quoted as simply the time in convenient units (ms or milliseconds in this case). ] Identify an event in your life (such as receiving a paycheck) that occurs regularly. Identify both the period and frequency of this event. #solutionbox[ I visit my parents for dinner every other Sunday. The frequency of my visits is 26 per calendar year. The period is two weeks. ] === Test Prep for AP Courses A mass attached to a spring oscillates and completes 50 full cycles in 30 s. What is the time period and frequency of this system? #solutionbox[ The frequency is given by #math.equation(block: true, alt: "f equals the fraction 1 over T equals the fraction 50 c y c l e s over 30 s equals 1.66 H z")[$f " " = " " frac(1, T) " " = " " frac(50 " " c y c l e s, 30 s) " " = " " 1.66 " " H z$] Time period is: #math.equation(block: true, alt: "T equals the fraction 1 over f equals the fraction 1 over 1.66 equals 0.6 s")[$T " " = " " frac(1, f) " " = " " frac(1, 1.66) " " = " " 0.6 " " "s"$] ] === Section Summary - Periodic motion is a repetitious oscillation. - The time for one oscillation is the period #math.equation(block: false, alt: "T")[$T$]. - The number of oscillations per unit time is the frequency #emph[#math.equation(block: false, alt: "f")[$f$]]. - These quantities are related by #math.equation(block: true, alt: "f equals the fraction 1 over T .")[$f = frac(1, T) .$] === Problems & Exercises What is the period of #math.equation(block: false, alt: "60 . 0 Hz")[$"60" "." 0 #h(0.25em) "Hz"$] electrical power? #solutionbox[ 16.7 ms ] If your heart rate is 150 beats per minute during strenuous exercise, what is the time per beat in units of seconds? #solutionbox[ #math.equation(block: true, alt: "0.400 s / beats")[$0.400 s / "beats"$] ] Find the frequency of a tuning fork that takes #math.equation(block: false, alt: "2 . 50 times 10 to the power minus 3 s")[$2 "." "50" × "10"^(− 3) #h(0.25em) "s"$] to complete one oscillation. #solutionbox[ 400 Hz ] A stroboscope is set to flash every #math.equation(block: false, alt: "8 . 00 times 10 to the power minus 5 s")[$8 "." "00" × "10"^(− 5) #h(0.25em) "s"$]. What is the frequency of the flashes? #solutionbox[ 12,500 Hz ] A tire has a tread pattern with a crevice every 2.00 cm. Each crevice makes a single vibration as the tire moves. What is the frequency of these vibrations if the car moves at 30.0 m/s? #solutionbox[ 1.50 kHz ] Engineering Application Each piston of an engine makes a sharp sound every other revolution of the engine. (a) How fast is a race car going if its eight-cylinder engine emits a sound of frequency 750 Hz, given that the engine makes 2000 revolutions per kilometer? (b) At how many revolutions per minute is the engine rotating? #solutionbox[ (a) 93.8 m/s (b) #math.equation(block: false, alt: "11 . 3 times 10 cubed rev/min")[$" 11 " " . " 3 × attach(" 10 ", t: 3) #h(0.25em) " rev/min "$] ]