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AP Unit 14 Problems

21 learning problems for Unit 14. Draw the situation, name the model and explain why each step follows. These are authored practice, with worked solutions on a separate page.

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Problem 1

Topic 14.1 · medium · mixed

A rope pulse travels right past a painted mark; a loudspeaker makes a continuous tone in still air.

Part a

Describe the painted mark’s motion and compare a single pulse with the tone. Identify each wave’s medium and disturbance direction.

Part b

The tone initially has equilibrium pressure 101325 Pa, peak pressure 101328 Pa, and minimum pressure 101322 Pa. Determine its pressure amplitude. That amplitude then doubles at unchanged frequency and medium. Explain how loudness and energy transfer change; then predict how warming the air affects speed.

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Problem 2

Topic 14.1 · medium · experimental

Two 3.0 m strings each have tension 16 N. String P has mass 0.012 kg and string Q has mass 0.048 kg.

Part a

Find each small-pulse speed and travel time.

Part b

Design a comparison to test the tension dependence while keeping mass per length controlled. What graph would support the model?

Part c

Compare what fixes string speed, sound speed in warmer air, and radio versus visible speed in vacuum.

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Problem 3

Topic 14.2 · medium · graphical

A snapshot of a sinusoidal wave at t = 0 has crests at x = 0.10 m and 0.90 m; its displacement extremes are ±0.015 m. A fixed-point trace repeats every 0.20 s.

Part a

Determine A, λ, T, f, and v. Write a possible fixed-point equation starting at a crest.

Part b

Construct labeled sketches of y versus x and y versus t. Explain why a vertical rescaling alone would not change f or λ.

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Problem 4

Topic 14.2 · medium · mixed

Two generators drive identical ideal strings at equal displacement amplitude. Generator R has 6 Hz; generator S has 12 Hz. The strings carry waves at 18 m/s.

Part a

Compare periods, wavelengths, and energy transfer; state the model conditions behind your comparison.

Part b

For a separate sound comparison, R sounds at 480 Hz and S at 960 Hz, preserving the 2:1 ratio but using ordinary audible tones. Which has higher pitch? Could R be made louder without raising its pitch?

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Problem 5

Topic 14.3 · medium · mixed

A 20 Hz wave travels from a string with μ = 0.020 kg/m to one with μ = 0.0050 kg/m; both have tension 8.0 N.

Part a

Find speeds and wavelengths, and predict the orientation of a reflected upward pulse.

Part b

A student claims no transmitted wave can exist if a reflected pulse is observed. Evaluate the claim using energy accounting.

Part c

Reverse the source: an upward pulse now travels from the lighter, faster string into the heavier, slower string at the same tension. Predict its reflected displacement orientation and compare the reflected and transmitted travel speeds.

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Problem 6

Topic 14.3 · medium · mixed

Unpolarized light of intensity 80 W/m² crosses an ideal vertical polarizer and then an ideal analyzer at 60° to vertical. The final beam illuminates 0.015 m².

Part a

Calculate the final intensity and average transmitted power. Explain the averaging in the definition of intensity.

Part b

Rotate the analyzer to horizontal and compare with trying to polarize air sound using crossed slots.

Part c

After reflection from glass, a beam’s transmitted intensity through a rotating analyzer varies. Explain why this is compatible with transverse light. Name two other interactions that may polarize transverse waves, and explain why this observation does not imply longitudinal sound can be polarized.

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Problem 7

Topic 14.4 · medium · mixed

A vacuum beam has wavelength 0.030 m and travels along +x. Its E field points along +z at one instant.

Part a

Calculate f and draw mutually perpendicular E, B, and propagation directions, including a plane wavefront.

Part b

Would evacuating the beam path stop propagation? Explain using the oscillating physical quantities.

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Problem 8

Topic 14.4 · medium · mixed

An observatory detects radio, microwave, infrared, visible, ultraviolet, X-ray, and gamma radiation from a source.

Part a

Arrange all seven in decreasing wavelength. Within visible light give the corresponding color order.

Part b

Two simultaneous pulses travel the same vacuum distance, one radio and one gamma. Compare travel time and frequency; explain why an exact category cutoff is unnecessary. Explain the narrower and broader uses of the word light.

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Problem 9

Topic 14.5 · medium · mixed

A subsonic ambulance passes a stationary listener in still air. The siren’s source frequency stays constant.

Part a

Draw equal-time emitted wavefronts just before and just after passage and predict the observed frequency relative to the source.

Part b

Repeat at a greater subsonic ambulance speed along the same path. Compare the shifts and explain why increasing loudness alone would not do this.

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Problem 10

Topic 14.5 · medium · mixed

A siren and listener move to the right in still air, both at subsonic speeds relative to the air, with the listener ahead of the siren. Consider three cases: same velocity; siren faster; listener faster.

Part a

Rank observed frequency relative to the rest frequency in the three cases, using front spacing and encounter rate.

Part b

Could two different source–observer configurations with the same closing speed be guaranteed to produce the same sound shift? Explain the limitation of that shortcut.

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Problem 11

Topic 14.6 · medium · graphical

At t = 0 two pulses overlap. At positions x = 0,1,2,3,4 cm their displacements are P = [0,2,4,2,0] mm and Q = [0,−2,−4,−2,0] mm.

Part a

Construct the net snapshot and explain what occurs after the pulses pass through each other.

Part b

If Q has half its stated amplitude, construct the new net graph and decide whether the interference is still destructive.

Part c

Instead make Q an upward pulse with half of P’s amplitude at each listed position. Construct the net snapshot, classify the overlap, and explain what changes in the resultant amplitude.

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Problem 12

Topic 14.6 · medium · mixed

A 0.60 m string fixed at both ends has wave speed 120 m/s. A tuning fork pair produces tones at 500 Hz and 504 Hz.

Part a

Sketch the string’s fundamental and third harmonic with labeled nodes/antinodes. Calculate their wavelengths and frequencies; compare with a same-length closed–open pipe at the same speed.

Part b

Find the fork beat frequency and explain the amplitude variation in terms of relative phase.

Part c

Replace the string’s right fixed end with an ideal loose end. Identify the new end conditions and fundamental frequency. Compare with a 0.60 m pipe open at both ends at the same 120 m/s wave speed. Finally, compare an ideal string loose at both ends with that open–open pipe.

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Problem 13

Topic 14.7 · medium · mixed

For a single slit, λ = 500 nm, a = 0.10 mm, and screen distance L = 1.0 m.

Part a

Find the first two minima on both sides and central bright width. Check the angular approximation. Interpret the supplied dimensionless intensity plot: identify the central maximum, its bounding dark orders, and why its interval is wider than a side lobe.

A single-slit intensity pattern. The single-slit curve is plotted versus u=a sinθ/λ. The central maximum is at u=0, bounded by dark minima at −1 and +1. Further dark minima occur at nonzero integers ±2, ±3, ±4; side lobes are lower and occupy intervals of width one in u, while the central bright interval has width two. There is no dark minimum at zero.

Part b

Explain the cancellation at the first minimum by pairing locations across the opening.

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Problem 14

Topic 14.7 · medium · experimental

A class compares diffraction through a long narrow slit, a circular pinhole, and a doorway using visible light and sound.

Part a

Predict qualitative spreading and pattern shapes. State the opening-to-wavelength comparison supporting each claim.

Part b

Design a visible-light test for how slit width affects a diffraction pattern, giving controls, measurements, and an uncertainty strategy.

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Problem 15

Topic 14.8 · medium · mixed

Two slits each have width a = 0.10 mm and separation d = 0.40 mm. Light wavelength is 600 nm and L = 2.0 m.

Part a

Find fine bright-fringe spacing and the first single-slit envelope minimum. Which bright interference order is missing first?

Part b

Sketch the expected pattern and explain why observing these fringes supports a wave model of light.

Part c

For an interference-only double-slit model, label separation d and far-field angle θ on a path diagram, then derive the bright-order screen relation at angles below 10°. State the path difference for a dark interference direction.

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Problem 16

Topic 14.8 · medium · mixed

White light falls on a grating with 400 lines/mm. Consider red λ = 650 nm and violet λ = 400 nm.

Part a

Find first-order angles and explain the central maximum’s color.

Part b

Determine each color’s largest possible order. Would adding more slits at the same d move these principal directions?

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Problem 17

Topic 14.9 · medium · mixed

A soap film (n = 1.33) in air is illuminated normally with vacuum wavelength 532 nm. Absorption is negligible.

Part a

Label the phase change at each reflection and find the smallest positive thickness for constructive reflected phase. State which portions of the incoming light can leave each interface and whether absorption is present in this lossless model.

Part b

Find the next positive thickness for destructive reflected phase, and explain why a changing-thickness film changes color. For an oil film of index 1.20 on water of index 1.33 in air, compare its reflection reversals with the soap film and explain how thickness variation can still select reflected colors; no additional thickness calculation is required.

Part c

At a different, absorptive interface, 100 mW of incident optical power gives 12 mW reflected power and 68 mW transmitted power. Determine absorbed power and explain the energy partition.

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Problem 18

Topic 14.9 · medium · mixed

An antireflection coating with n = 1.40 is placed on glass of n = 1.96, surrounded by air. Its design vacuum wavelength is 560 nm at normal incidence.

Part a

Find the minimum coating thickness and justify both the phase and amplitude conditions for ideal cancellation. Does refraction into or out of the coating add the same abrupt π reversal as reflection?

Part b

Explain what happens if the wavelength or incidence angle changes, and identify the AP quantitative scope.

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Problem 19

Topic 14.1 Topic 14.2 Topic 14.6 · hard · experimental

A class measures standing waves on a uniform 1.00 m string fixed at both ends. Its mass is 0.0100 kg. Fundamental frequencies for tensions 4.0, 9.0, and 16.0 N are 10.0, 15.0, and 20.0 Hz.

Part a

Construct a graph with labeled axes, units, and scales to test the string-speed and standing-wave models together, find its predicted slope, and compare the data.

Part b

Describe how to locate resonance reliably, estimate uncertainty, and detect a boundary that behaves partly loose rather than fixed.

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Problem 20

Topic 14.2 Topic 14.7 Topic 14.8 · hard · experimental

A double-slit experiment gives bright maxima at y = 0, ±3.0, ±6.0 mm, with L = 1.20 m and d = 0.20 mm. Its first envelope zeros occur at ±12 mm.

Part a

Infer the wavelength and slit width. Identify the missing order and compare the frequency with light of wavelength 650 nm.

Part b

A center-placement error shifts every measured y by +0.5 mm. Give a method to reduce this systematic effect and explain which inferred quantities are vulnerable.

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Problem 21

Topic 14.2 Topic 14.3 Topic 14.8 Topic 14.9 · hard · mixed

A 600 nm vacuum laser crosses a polarizer, then a normal-incidence coating (n = 1.50) on glass (n = 2.25), and is analyzed with a grating of spacing 2.0 μm in air.

Part a

Find the light’s frequency, wavelength in the coating, and minimum ideal antireflection thickness. Explain which quantities a stationary interface or polarizer changes.

Part b

Calculate the first grating order angle and decide whether rotating the polarizer alone should move it in the ideal setup.

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More chapter problems

Try these problems before opening the solutions. Show the physical model, your calculation or sketch, and the reason your conclusion follows.

Identify the repeating model from observations

A speaker emits a sustained pure tone in still air. A fixed pressure sensor records equal-phase maxima every 0.0100 s. A simultaneous spatial pressure map shows successive compressions 3.40 m apart. The speaker is then switched off and makes one short isolated click.

(a) State the appropriate continuous-wave model for the sustained tone, explicitly identify its well-defined wavelength and frequency from the two observations, and calculate propagation speed. Explain why the fixed-location measurement and snapshot cannot exchange their horizontal units.

14.1.A.1.ii 14.2.A.1.ii 14.2.A.1.vi 14.2.A.3

(b) Compare the isolated click with the sustained tone: state which is a pulse and why assigning the click one repeating crest spacing and cycle rate is not justified by these observations. Describe local air-particle motion relative to propagation and explain why sound does not carry those same particles from the speaker to the sensor.

14.1.A.1 14.1.A.1.i 14.1.A.5 14.1.A.5.i

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Worked solution and rubric →Revisit the explanation →Revisit the explanation →Read the relevant lesson →

Write and draw two views of the same traveling wave

A sinusoidal wave travels right on a uniform ideal string. Displacement extremes are ±3.00 mm. Adjacent crests are 1.20 m apart; a marked segment repeats every 0.400 s. At t = 0 a crest is at x = 0.300 m. Use x in metres, t in seconds and y in millimetres.

(a) Determine A, λ, T, f and v. Write a fixed-location displacement-time equation at x = 0.300 m starting at a crest, and a fixed-time displacement-position equation at t = 0.100 s. Keep the given phase origin and rightward direction consistent.

14.2.A.2 14.2.A.3 14.2.A.1.i 14.2.A.1.ii 14.2.A.1.vi

(b) Create the actual y-versus-t graph at x = 0.300 m from 0 to 0.800 s and the actual y-versus-x snapshot at t = 0.100 s from 0 to 2.40 m. Label axes, numerical scales, units, extrema and zero crossings. Locate the first nonnegative spatial crest on the snapshot and use its shift from t = 0 to check v.

(c) The source doubles displacement amplitude on the unchanged ideal string but keeps its frequency fixed. Predict changes in both graphs, their horizontal repeats, and wave speed. Then explain why an antinode of a later standing wave would not mean a point stays at maximum displacement.

14.2.A.1.iii 14.6.B.1.i

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Worked solution and rubric →Revisit the explanation →Revisit the explanation →Read the relevant lesson →

Build a standing wave instead of assuming one

A 1.50 m ideal string is fixed at both ends. Two steady sinusoidal traveling contributions confined to the string are yᵣ = (1.50 mm) sin[2πx/(1.00 m) − 80πt] and yₗ = (1.50 mm) sin[2πx/(1.00 m) + 80πt], with t in seconds. Use sin(a−b)+sin(a+b) = 2 sin a cos b. The separate waves propagate in opposite directions.

(a) Add the displacements to obtain y(x,t), and explain why two waves traveling in opposite directions within the confined region can form a standing pattern. Verify that the sum satisfies both fixed endpoints at every time, even though each separate contribution need not vanish at an endpoint.

14.6.B.1 14.6.B.1.ii

(b) Draw the separate right-moving and left-moving profiles and their pointwise sum over the full string at t = 0, T/4 and T/2. Use a common vertical scale in millimetres and horizontal scale in metres. Compute the three times, label permanent nodes/antinodes, and explain why the all-zero snapshot at T/4 does not make every point a node.

14.6.B.1 14.6.B.1.i 14.6.B.3

(c) Use the diagram’s node spacing to determine λ. Calculate v, identify the harmonic number, and determine the fundamental frequency. Explain why changing the right endpoint from fixed to ideally loose changes the allowed standing patterns.

14.6.B.1.ii 14.6.B.2 14.6.B.3

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Worked solution and rubric →Revisit the explanation →Revisit the explanation →Read the relevant lesson →

Diagnose a time axis mistaken for a spatial axis

At x = 0, a string wave is y(t) = (2.00 mm) cos(20πt), with t in seconds. A t = 0 snapshot is y(x) = (2.00 mm) cos(4πx), with x in metres.

(a) Which interpretation is consistent with both equations?

14.2.A.2 14.2.A.3

Worked solution and rubric →Revisit the explanation →Revisit the explanation →Read the relevant lesson →

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