Measure Wave Amplitude

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Middle School Physical Science › Measure Wave Amplitude

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1

A water wave passes by a dock. The water surface at rest is the equilibrium position. The highest point of the wave is $5\text{ cm}$ above the calm surface and the lowest point is $5\text{ cm}$ below it. What is the amplitude of the wave?

$10\text{ cm}$

$2.5\text{ cm}$

$-5\text{ cm}$

$5\text{ cm}$

Explanation

This question tests understanding that amplitude is the maximum displacement from the rest (equilibrium) position to the crest or trough, measured as a distance, not the total wave height from crest to trough. Amplitude is defined as the maximum distance a wave moves from its rest/equilibrium position: for a water wave, the rest position is the calm surface, the crest is the highest point (5 cm above calm), and the amplitude is the distance from rest to crest (A=5 cm). The trough (5 cm below calm) also gives A=5 cm magnitude, confirming consistency for symmetric waves. Choice B is correct because it measures amplitude as 5 cm, the maximum displacement from the equilibrium calm surface to the crest or trough. Choice A is wrong because it adds crest and trough for total height (10 cm=2A); Choice C halves incorrectly (2.5 cm); Choice D uses negative when amplitude is positive. Measuring wave amplitude systematically: (1) locate rest/equilibrium (calm surface), (2) identify maximum point (crest 5 cm above), (3) measure distance (5 cm), (4) record A=5 cm, and (5) verify with trough (5 cm below, same magnitude). Larger amplitude means taller wave: 5 cm amplitude is taller than 2 cm, carrying (5/2)²=6.25× more energy, emphasizing correct measurement to avoid mistaking total height for amplitude.

2

A pendulum’s equilibrium position is straight down at $\theta=0^\circ$. It swings to a maximum of $15^\circ$ to the left and $15^\circ$ to the right. What is the amplitude of the pendulum’s motion?

$0^\circ$

$-15^\circ$

$15^\circ$

$30^\circ$ (left to right)

Explanation

This question tests understanding that amplitude is the maximum displacement from the rest (equilibrium) position to the crest or trough, measured as a distance, not the total wave height from crest to trough. Amplitude is defined as the maximum distance a wave moves from its rest/equilibrium position: for a pendulum, rest is θ=0°, max swing to 15° (left or right), giving A=15°. Like a transverse wave, the 'crest' is max angle one side (+15°), trough the other (-15°), but amplitude is the max displacement magnitude (15°). Choice B is correct because it measures the maximum angular displacement from equilibrium (15°), treating left and right symmetrically. Choice A adds total swing (30°=2A); Choice C is equilibrium itself; Choice D is negative. Measuring wave amplitude systematically: (1) locate equilibrium (0°), (2) identify max displacement (15°), (3) measure angle (15°), (4) record A=15°, and (5) verify both sides same. Amplitude significance in pendulums: larger A means wider swing, more energy, preventing confusion of total arc with amplitude.

3

A student measures the vertical distance from the crest of a wave to the trough and gets $8\text{ cm}$. The wave is symmetric about its equilibrium (rest) position. What is the amplitude?

$16\text{ cm}$

$-4\text{ cm}$

$4\text{ cm}$

$8\text{ cm}$

Explanation

This question tests understanding that amplitude is the maximum displacement from the rest (equilibrium) position to the crest or trough, measured as a distance, not the total wave height from crest to trough. Amplitude is defined as the maximum distance a wave moves from its rest/equilibrium position: if crest-to-trough is 8 cm and symmetric, then A=4 cm (half the total height). For symmetric waves, the total wave height (crest to trough) is 2A, so measuring 8 cm gives A=8/2=4 cm. Choice B is correct because it recognizes the given crest-to-trough distance as 2A and halves it for amplitude (4 cm). Choice A uses total height as A; Choice C doubles incorrectly; Choice D is negative. Measuring wave amplitude systematically: (1) confirm symmetry about equilibrium, (2) measure total height (8 cm), (3) divide by 2 (4 cm), (4) record A=4 cm, and (5) verify consistency. This method is useful when only total height is given, preventing the common mistake of equating total height with amplitude.

4

A wave is drawn on grid paper. The equilibrium line is the middle horizontal line labeled $y=0\text{ cm}$. Each grid square represents $1\text{ cm}$ vertically. The crest is 4 squares above the equilibrium line. What is the amplitude?

4 cm

4 squares (no units)

2 cm

8 cm

Explanation

This question tests understanding that amplitude is the maximum displacement from the rest (equilibrium) position to the crest or trough, measured as a distance, not the total wave height from crest to trough. Amplitude is defined as the maximum distance a wave moves from its rest/equilibrium position: on grid paper, rest is the middle line at y=0 cm, the crest is 4 squares above (each 1 cm, so +4 cm), and amplitude is the distance from rest to crest (4 cm)—if there's a trough, it would confirm, but crest measurement suffices. A common mistake is measuring from some other reference or forgetting the scale, like counting squares without units or doubling for total height. Choice B is correct because it properly uses the scale (4 squares × 1 cm/square = 4 cm) to measure displacement from equilibrium to crest, accurately identifying rest at the middle line. Choice A might double it to 8 cm thinking crest-to-trough; Choice C halves to 2 cm incorrectly; Choice D omits units, just giving squares, but amplitude needs units like cm. Measuring wave amplitude systematically: (1) locate rest (middle line y=0), (2) identify max point (crest 4 squares up), (3) measure using scale (4 × 1 cm = 4 cm), (4) record A=4 cm, and (5) verify if trough shown (would be same if symmetric). Amplitude significance: 4 cm amplitude indicates energy level (E ∝ A²), useful in contexts like rope waves where larger A means more vigorous shaking.

5

Two waves are compared using the same vertical scale (in cm) with the rest position at $y=0$. Wave A has a crest at $+2\text{ cm}$ and a trough at $-2\text{ cm}$. Wave B has a crest at $+5\text{ cm}$ and a trough at $-5\text{ cm}$. Which statement is correct?

Both waves have the same amplitude because both cross the rest position.

Wave A has larger amplitude because 2 cm is greater than 5 cm.

Wave B has larger amplitude because 5 cm is greater than 2 cm.

Neither wave has an amplitude because amplitude is measured crest-to-trough.

Explanation

This question tests understanding that amplitude is the maximum displacement from the rest (equilibrium) position to the crest or trough, measured as a distance, not the total wave height from crest to trough. Amplitude is defined as the maximum distance a wave moves from its rest/equilibrium position: both waves have rest at y=0, Wave A has crest at +2 cm (A=2 cm) and trough at -2 cm, while Wave B has crest at +5 cm (A=5 cm) and trough at -5 cm, so Wave B's amplitude (5 cm) is larger than Wave A's (2 cm) since 5 > 2. A common mistake is measuring from crest to trough and comparing totals (Wave A: 4 cm, Wave B: 10 cm), but amplitude is half that for symmetric waves—still Wave B larger—or reversing which is bigger. Choice B is correct because it properly identifies Wave B's amplitude as 5 cm (rest to +5 cm) which is greater than Wave A's 2 cm (rest to +2 cm), using the same scale and rest position. Choice A reverses the comparison, claiming 2 cm > 5 cm incorrectly; Choice C assumes same amplitude because both cross rest, ignoring actual displacements; Choice D wrongly states neither has amplitude, confusing with crest-to-trough measurement which is actually 2A. Measuring wave amplitude systematically: (1) locate rest (y=0), (2) identify max displacement (Wave A: 2 cm, Wave B: 5 cm), (3) measure from rest, (4) record A, and (5) compare magnitudes (5 cm > 2 cm). Amplitude significance: Wave B has larger amplitude, thus more energy (E ∝ A², (5/2)²=6.25× more than A), making it 'taller' and more intense.

6

A student measures the vertical distance from the crest of a wave to the trough and gets 12 cm. The rest position is halfway between crest and trough. What is the amplitude of the wave?

6 cm

-6 cm

3 cm

12 cm

Explanation

This question tests understanding that amplitude is the maximum displacement from the rest (equilibrium) position to the crest or trough, measured as a distance, not the total wave height from crest to trough. Amplitude is defined as the maximum distance a wave moves from its rest/equilibrium position: if crest-to-trough is 12 cm and rest is halfway, then from rest to crest is 6 cm (half of 12 cm) and rest to trough is 6 cm, so amplitude is 6 cm—assuming symmetry where rest is midway. A common mistake is taking the full 12 cm as amplitude instead of half, or halving incorrectly to 3 cm, or using negative values. Choice B is correct because it recognizes that with rest halfway, amplitude is half the crest-to-trough distance (12 cm / 2 = 6 cm), accurately measuring maximum displacement from equilibrium. Choice A uses the full 12 cm, confusing with 2A; Choice C quarters to 3 cm mistakenly; Choice D uses negative (-6 cm) when amplitude is positive magnitude. Measuring wave amplitude systematically: (1) locate rest (halfway between crest and trough), (2) identify max point (crest or trough), (3) measure distance from rest (6 cm), (4) record A=6 cm, and (5) verify (total height 12 cm = 2A, yes). Amplitude significance: 6 cm amplitude means more energy (E ∝ A²) than smaller waves, important for understanding wave impact like in oceans.

7

A pendulum swings back and forth. Its equilibrium position is straight down at $\theta=0^\circ$. The pendulum reaches a maximum angle of $15^\circ$ to the left and $15^\circ$ to the right. What is the amplitude of the pendulum’s motion?

15°

-15°

30°

Explanation

This question tests understanding that amplitude is the maximum displacement from the rest (equilibrium) position to the crest or trough, measured as a distance, not the total wave height from crest to trough. Amplitude is defined as the maximum distance a wave moves from its rest/equilibrium position: for a pendulum, the rest position is straight down at θ=0°, the maximum to the right is +15° (one peak), and the amplitude is the distance from rest to that max (15°); the left max at -15° is also 15° from rest (magnitude), confirming amplitude is 15°—it's the same on both sides due to symmetry. A common mistake is measuring from left to right (-15° to +15° = 30° total), but this gives twice the amplitude (2A), not the amplitude itself—amplitude is always measured from the rest position to the peak, not peak to peak. Choice B is correct because it accurately measures amplitude as the maximum angular displacement from equilibrium (0°) to the peak (15° to left or right), using the magnitude. Choice A measures total swing (30°) calling it amplitude when that's 2A; Choice C uses negative (-15°) when amplitude is positive; Choice D assumes no amplitude at rest, ignoring max deviation. Measuring wave amplitude systematically: (1) locate rest (θ=0°), (2) identify max displacement (15° left or right), (3) measure from rest (15°), (4) record A=15°, and (5) verify both sides equal. Amplitude significance: larger amplitude means wider swing and more energy (E ∝ A²), affecting pendulum motion like period approximation for small angles.

8

A water wave moves across a pool. The calm water level is the equilibrium (rest) position. At one moment, the crest is 5 cm above the calm level and the trough is 5 cm below the calm level. What is the wave’s amplitude?

2.5 cm

5 cm

10 cm

-5 cm

Explanation

This question tests understanding that amplitude is the maximum displacement from the rest (equilibrium) position to the crest or trough, measured as a distance, not the total wave height from crest to trough. Amplitude is defined as the maximum distance a wave moves from its rest/equilibrium position: for a water wave, the rest position is the calm water level, the crest is 5 cm above calm (maximum positive displacement), and the amplitude is the distance from rest to crest (5 cm); the trough 5 cm below calm is also 5 cm from rest (magnitude), confirming amplitude is 5 cm—it's the same whether measured to crest or to trough because the wave is symmetric. A common mistake is measuring from crest to trough (5 + 5 = 10 cm total), but this gives twice the amplitude (2A), not the amplitude itself—amplitude is always measured from the rest position to the peak, not peak to peak. Choice B is correct because it accurately measures amplitude as the distance from the calm level (rest) to the crest (5 cm above) or to the trough (5 cm below), using the magnitude of the maximum displacement. Choice A measures crest to trough (10 cm) calling it amplitude when that's 2A, and amplitude is 5 cm; Choice C uses a negative value (-5 cm) when amplitude is positive (5 cm); Choice D halves it incorrectly to 2.5 cm, perhaps confusing with average. Measuring wave amplitude systematically: (1) locate rest/equilibrium position (calm water level), (2) identify maximum displacement point (crest 5 cm above), (3) measure distance from rest to that point (5 cm), (4) record as amplitude with units (A=5 cm), and (5) verify with trough (also 5 cm). Amplitude significance: larger amplitude means taller wave and more energy (E ∝ A², so 5 cm amplitude has (5/2)²=6.25× more energy than 2 cm), affecting physical impact like wave power.

9

A student measures a wave and says: “The crest is at $+4\text{ cm}$ and the trough is at $-4\text{ cm}$, so the amplitude is $8\text{ cm}$.” Which response correctly fixes the mistake?

The student is correct: amplitude is always crest-to-trough.

Amplitude is measured from equilibrium to a crest (or trough), so the amplitude is $4\text{ cm}$.

Amplitude is the horizontal distance from one crest to the next.

Amplitude should be negative for troughs, so the amplitude is $-4\text{ cm}$.

Explanation

This question tests understanding that amplitude is the maximum displacement from the rest (equilibrium) position to the crest or trough, measured as a distance, not the total wave height from crest to trough. Amplitude is defined as the maximum distance a wave moves from its rest/equilibrium position: the student's measurement of +4 cm to -4 cm totals 8 cm, which is 2A, but actual amplitude is 4 cm from rest to crest. The wave is symmetric, so trough also gives 4 cm magnitude; a common mistake is measuring crest-to-trough as amplitude, but amplitude is rest-to-peak. Choice C is correct because it explains amplitude is measured from equilibrium to crest (or trough), giving 4 cm. Choice A agrees with the mistake (crest-to-trough as amplitude); Choice B suggests negative for troughs incorrectly (amplitude positive); Choice D confuses with wavelength (horizontal distance). Measuring wave amplitude systematically: (1) locate equilibrium (center), (2) identify crest (+4 cm), (3) measure from rest (4 cm), (4) record A=4 cm, and (5) verify with trough (4 cm). Correcting this prevents overestimating energy (E ∝ A²: true A=4 cm has 16× for 1 cm, but mistaken 8 cm would inflate to 64×).

10

A wave is drawn on grid paper where each vertical grid square represents $1\text{ cm}$. The equilibrium line is the middle horizontal line. A crest is 4 squares above equilibrium and a trough is 4 squares below equilibrium. What is the amplitude of the wave?

$2\text{ cm}$

$-4\text{ cm}$

$4\text{ cm}$

$8\text{ cm}$

Explanation

This question tests understanding that amplitude is the maximum displacement from the rest (equilibrium) position to the crest or trough, measured as a distance, not the total wave height from crest to trough. Amplitude is defined as the maximum distance a wave moves from its rest/equilibrium position: for a wave on grid paper, the rest position is the middle horizontal line, a crest 4 squares above means maximum positive displacement, and with each square 1 cm, amplitude is 4 cm from rest to crest. The trough 4 squares below is also 4 cm from rest (magnitude), confirming amplitude is 4 cm—it's the same whether measured to crest or to trough because waves are typically symmetric about equilibrium; a common mistake is measuring from crest to trough (8 squares = 8 cm total), but this gives twice the amplitude (2A), not the amplitude itself—amplitude is always measured from the rest position to the peak, not peak to peak. Choice B is correct because it properly uses the scale to measure displacement (4 squares × 1 cm/square = 4 cm) from the equilibrium line to the crest (or trough). Choice A measures crest to trough (8 cm) calling it amplitude when that's 2A, amplitude is 4 cm; Choice C halves incorrectly to 2 cm; Choice D uses a negative value (-4 cm) when amplitude is positive magnitude. Measuring wave amplitude systematically: (1) locate rest/equilibrium position (middle horizontal line), (2) identify maximum displacement point (crest 4 squares above), (3) measure distance (count squares: 4 × 1 cm = 4 cm), (4) record as amplitude (A = 4 cm), and (5) verify with trough (also 4 cm). Common measurement contexts: on grid paper, ensure scale is applied correctly (here 1 cm/square), and amplitude indicates energy (E ∝ A²: 4 cm has 16× more than 1 cm), preventing errors like miscounting squares or confusing with total height.

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