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Flashcards in Periodic Motion & Waves Deck (61)
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1
Q

Where is the amplitude of the waveform shown below?

A

Amplitude is the distance between the average value and the most extreme value of the waveform.

Note that the amplitude is not the full range from minimum to maximum; that value is actually twice the amplitude.

2
Q

What is the amplitude of the waveform below, if each gridline division represents 1 cm?

A

The amplitude of the wave is 3 cm.

Note that the amplitude is not the full range from minimum to maximum; that value (6 cm in this example) is actually twice the amplitude.

3
Q

What is the period of the waveform shown below?

A

The period (T) is the amount of time it takes the wave to complete one full oscillation.

Note that the period is not the range from one zero-displacement position to the next; that value only captures one-half of the oscillation, and represents one-half of the period.

4
Q

What is the period of the waveform below, if each horizontal division represents 1 s?

A

The period of the wave is 6.5 s.

Note that the period is not the range between adjacent zero-displacement positions; that value (slightly over 3 s in this example) is actually one-half of the period.

5
Q

The period of Waveform 1 is twice that of Waveform 2. Which wave is oscillating more rapidly?

A

Waveform 2 is oscillating more rapidly.

The period is the time needed for the waveform to complete one full oscillation. The larger the period, the more time it takes for an oscillation to complete, and the slower the oscillation.

6
Q

What is the frequency of a waveform?

A

The frequency of a waveform is a measure of how rapidly the waveform oscillates.

Specifically, a wave’s frequency is the number of cycles, or total waves, that cross a certain point within a certain timespan.

7
Q

What units are used to measure frequency?

A

The units generally used to measure frequency are hertz (Hz)

The units for hertz are 1/s, which can be thought of as cycles or revolutions per second.

8
Q

What is the mathematical relationship between a wave’s frequency and its period?

A

The frequency of a waveform is calculated as f = 1/T.

Here, f is the waveform’s frequency in Hz, while T is its period in s.

9
Q

How is the angular frequency (ω) of an oscillating system calculated?

A

The angular frequency (ω) is calculated as:

ω = 2πf

The units of ω are radians/s. Angular frequency is used in reference to rotating objects, such as a gear or rolling tire.

10
Q

What is the frequency of the waveform shown below, if each horizontal division represents 1 second?

A

The waveform’s frequency is 0.2 Hz.

The period of the waveform is 5 s, and the frequency f = 1/T, or 1/5 s-1.

11
Q

Which waveform is oscillating more rapidly, if the frequency of waveform 1 is100 Hz, while that of waveform 2 is 200 Hz?

A

Waveform 2 is oscillating more rapidly.

Since frequency is inversely proportional to period, it has a direct relationship with wave oscillation speed; the higher a wave’s frequency, the more rapidly it oscillates.

12
Q

Define:

phase

This term has multiple meanings; here, define it as it relates to a waveform.

A

The phase of a waveform is the offset of the waveform relative to its origin, zero value.

On the MCAT, phase is usually tested as phase difference, or the discrepancy in phase between two distinct waves.

13
Q

What is the phase difference between two waveforms?

A

The phase difference between two waveforms is the value of the phase of the second waveform at the origin of the first waveform.

The picture above represents a phase difference of one-half of a wavelength. Notice that point A is the origin of the black waveform, while the red waveform is halfway through a full oscillation.

14
Q

The two waveforms below differ by one-half of a wavelength. What is the calculated phase difference in degrees and radians, respectively?

A

One-half of a wavelength corresponds to a phase difference of 180º, or Π radians.

To convert between degrees and radians, simply remember that a full cycle is 360º. That value is equal to 2Π radians.

15
Q

The two waveforms below differ by one-quarter of a wavelength. What is the calculated phase difference in degrees and radians, respectively?

A

One-quarter of a wavelength corresponds to a phase difference of 90º, or Π/2 radians.

To convert between degrees and radians, simply remember that a full cycle is 360º, which is equal to 2Π radians.

16
Q

What equation gives the relationship between the displacement of a spring and the force it generates?

A

Hooke’s Law, or Fx = -kx, relates force and displacement for a spring.

In this equation,

Fx = force exerted by the spring (N)
k = force constant of the spring (N/m)
x = extension or compression of the spring from equilibrium (m)

17
Q

The force constant of a vertically-hung spring is 1,000 N/m. By how much does it stretch when a 10-kg object is attached to it?

A

The spring stretches by 10 cm.

According to Hooke’s Law, Fx = -kx. The spring will stretch until the force equals the weight of the object, which is 100 N in this case. So:

100 = -(1,000)(x)
x = 10-1 m

18
Q

Define:

simple harmonic motion (SHM)

A

Simple harmonic motion is the motion produced when an object at equlibrium is displaced and feels a restorative force proportional to the displacement.

On the MCAT, the most common examples of SHM involve springs and pendulums.

19
Q

What is the shape of the graph that represents a simple harmonic motion system?

Assume that no frictional forces are present.

A

All simple harmonic motion systems have sinusoidal graphs.

For example, the above graph shows displacement vs. time for a mass on a spring.

20
Q

How does the graph of a mass on a spring change when the mass begins to oscillate more quickly?

A

The peaks of the graph will move closer together. In other words, the frequency will increase.

In the graph above, the red line represents a system which is oscillating more rapidly than the system shown by the black line.

21
Q

When does a system undergoing simple harmonic motion possess maximum kinetic energy?

A

The system has maximum kinetic energy when the object is at the equilibrium position.

Kinetic energy is proportional to the square of the velocity, and velocity is the slope of the line tangent to the position curve, shown above. The slope maximizes as the line crosses through equilibrium, and is at a minimum (0) at the extreme positions.

22
Q

When does a system undergoing simple harmonic motion possess maximum potential energy?

A

The system has maximum potential energy when the object is at the maximum distance from equilibrium, or at the amplitude.

Since the system’s total energy remains constant, the maximum potential energy occurs when the kinetic energy is at a minimum.

23
Q

What formula gives the frequency of an object oscillating on a spring?

A

ω = (k/m)½

Where:

k = force constant of the spring (N/m)
m = mass of the object (kg)

24
Q

Two objects are suspended from identical springs. Object 1 has a mass of 5 kg, while Object 2 has a mass of 10 kg. Which object oscillates more rapidly?

A

Object 1 oscillates at the higher frequency.

The frequency of an object oscillating on a spring is equal to (k/m)½. Since the frequency is inversely proportional to the square root of the mass, the larger the mass, the lower the frequency.

25
Q

Two identical objects are suspended from springs. Spring 1 has a force constant of 5,000 N/m, while Spring 2 has a force constant of 10,000 N/m. Which object oscillates more rapidly?

A

The mass on spring 2 oscillates at the higher frequency.

The frequency of an object oscillating on a spring is equal to (k/m)½. Since the frequency is directly proportional to the square root of the force constant, the larger the force constant, the higher the frequency.

26
Q

Two identical objects are suspended from identical springs. One is displaced 10 cm from equilibrium, while the other is displaced 20 cm from equilibrium. Which object oscillates more rapidly?

A

The two springs oscillate at the same frequency.

The frequency of an object oscillating on a spring is (k/m)½. The frequency depends only on the force constant of the spring and the mass of the object, not the displacement. In other words, frequency is independent of amplitude.

27
Q

What formula can be used to find the potential energy of a spring that is displaced from equilibrium?

A

The potential energy of a spring is given by U = ½kx2.

Here, k is the force constant of the spring; x is the displacement from equilibrium.

28
Q

What formula gives the maximum potential energy value of a mass oscillating on a spring?

A

Umax = ½kA2

The potential energy for any spring system is equal to ½kx2. The maximum value of x is A, the amplitude. When A is plugged in for x, the equation yields the maximum value of U.

29
Q

What formula can be used to find the kinetic energy of a mass oscillating on a spring?

A

The kinetic energy of a mass oscillating on a spring is given by KE = ½mv2.

Here, m is the mass of the oscillating object; v is the velocity of the object at the moment in question.

30
Q

What formula gives the maximum KE value for a mass oscillating on a spring?

A

½kA2

Since the energy of the system is constant, the maximum value of the kinetic energy is equal to the maximum value of the potential energy. At the equilibrium position, all the energy in the system is in the form of kinetic energy. In this equation, k is the spring constant and A is the amplitude.

31
Q

What formula gives the maximum velocity of a mass oscillating on a spring?

A

vmax = A(k/m)½ = Aω

The maximum value of the kinetic energy is ½kA2. Setting this equal to ½mv2 and solving for v yields the equation above.

32
Q

What formula gives the frequency of an oscillating pendulum?

A

f = (g/L)½

Where:

f = oscillation frequency (Hz)
g = gravitational acceleration (m/s2)
L = length of the pendulum (m)

33
Q

Pendulum 1 has a string four times the length of Pendulum 2. What is the difference between their oscillation frequencies?

A

Pendulum 2 has a frequency twice that of Pendulum 1.

From the equation f = (g/L)½, the frequency of a pendulum’s oscillation is inversely proportional to the square root of its length. Since the length of Pendulum 1 is 4 times that of Pendulum 2, its frequency differs by 1/√4; in other words, 1’s frequency is half of 2’s.

34
Q

How does a transverse wave propagate through a medium?

A

In a transverse wave, the oscillating particles are displaced perpendicular to the direction of wave propagation.

35
Q

What are some classic examples of transverse waves?

A

Transverse waves include:

  • light waves (electromagnetic waves)
  • string waves
  • stadium waves

Technically, water waves are classified as “surface waves” and are considered to be at least partially transverse. However, familiarity with this example is not necessary for the MCAT.

36
Q

How does a longitudinal wave propagate through a medium?

A

In a longitudinal wave, the oscillating particles are displacemed parallel to the direction of wave propagation.

37
Q

What are some classic examples of longitudinal waves?

A

Longitudinal waves include:

  • sound waves
  • spring waves (like that produced by a stretched Slinky)
38
Q

Calculate the period of the wave shown below:

A

The period equals 5 seconds.

A full wave cycle is any 360-degree segment and is usually easiest to picture as peak-to-peak or valley-to-valley. Notice that the wave valleys are present at exact intervals of 5 s.

39
Q

Calculate the frequency of the wave shown below:

A

The frequency equals 0.2 Hz.

Since frequency = 1/T, and the period is 5 seconds, f = 1/5 = 0.2.

40
Q

Calculate the amplitude of the wave shown below:

A

The amplitude equals 17 meters.

Amplitude is defined as the greatest positive displacement from the zero position of a wave.

41
Q

Calculate the wavelength of the wave shown below:

A

The wavelength equals 12.5 meters.

A full wave cycle is any peak-to-peak or valley-to-valley distance. Notice that the wave valleys are present at exactly 0 and 25, but this requires two wavelengths, not one. Hence, wavelength is 25 / 2, or 12.5 m.

42
Q

What is the speed of the wave below, if its wavelength is exactly 5 meters?

A

1 m/s

wave speed = (λ) (f) = (5) (1/5) = 1 m/s

43
Q

Define:

intensity

A

Intensity (here, in reference to sound) is the power per unit area of the wave.

The standard reference value for intensity is 10-12 W/m2.

44
Q

How does the intensity of a wave relate to its amplitude?

A

A wave’s intensity is proportional to the square of its amplitude:

I α A2

45
Q

When might constructive interference be observed?

A

Constructive interference occurs when two overlapping waves produce a new amplitude that is greater than that of either wave alone.

Maximum constructive interference occurs when the waves overlap with exactly the same speed, wavelength and frequency. This causes the final amplitude to be exactly the sum of the two original amplitudes.

46
Q

Two waves with amplitudes of 4m and 3m interfere, producing a wave with an amplitude of 7m. What type of interference occurred?

A

Constructive interference occurred. Specifically, this is the maximum amount of constructive interference possible with these two waves.

The resultant wave has an amplitude that is exactly the sum of the amplitudes of the two initial waves.

47
Q

When might destructive interference be observed?

A

Destructive interference occurs when two overlapping waves produce a new amplitude that is less than that of either wave alone.

Maximum destructive interference occurs when the waves are 180 degrees out of phase with exactly the same speed, wavelength and frequency. This causes the final amplitude to be exactly the difference of the two starting amplitudes.

48
Q

Two waves with amplitudes of 4m and 7m interfere, producing a wave with an amplitude of 3m. What type of interference occurred?

A

Destructive interference occurred. Specifically, this is the maximum amount of destructive interference possible with these two waves.

The resultant wave has an amplitude that is exactly the difference between the amplitudes of the two initial waves.

49
Q

Define:

resonance

This term has multiple meanings; here, define it as it relates to sound.

A

An oscillating system is undergoing resonance when it oscillates at its natural frequency. This tends to result in a dramatic increase in amplitude.

For example, a mass on a spring system resonates when an external force drives it at a frequency of (k/m)½.

50
Q

Define:

standing wave

A

A standing wave is one that exists at a fixed length in a given medium. Standing waves contain nodes and antinodes.

One example is a guitar string that is stationary at both ends, but has waves that propogate between the two ends. The diagram above shows a guitar string vibrating at the 4th harmonic.

51
Q

Define:

node

A

A node is the point on a standing wave that experiences no displacement. Nodes remain fixed in position.

In the image above, the nodes are shown by black dots.

52
Q

What condition must be met for a node to exist at the end of a standing wave?

A

The end where the node is produced must be fixed in position.

A guitar string tied down on both ends fits this condition; specifically, each end will display a node. An organ pipe with one end closed has a node at the closed end only.

53
Q

Define:

antinode

A

An antinode is the set of positions on a standing wave that experience the greatest displacement. A wave will fluctuate between antinode positions.

In the image above, the antinodes are shown by black dots.

54
Q

What condition must be met for an antinode to exist at the end of a standing wave?

A

The end where the antinode is produced must be open, not fixed.

A wind chime pipe with two open ends fits this condition; specifically, each end will display an antinode. An organ pipe with one end open has an antinode at the open end only.

55
Q

Define:

beat frequency

A

Beat frequency is the frequency of a wave that occurs due to the interference of two waves with slightly different frequencies.

Beat frequencies can be produced by any number of waves at once (3, 10, etc.), but the MCAT only tests beats with two waves.

56
Q

When two waves with frequencies f1 and f2 interfere, what equation gives the frequency of the beats?

A

The frequency of the beats is given by the absolute value of their difference: fbeat = |f1 - f2|.

57
Q

Tone A (220 Hz) and Tone B (224 Hz) are played simultaneously, creating a beat frequency. If A is increased by 12 Hz while B is increased by 4 Hz, how will the new beat frequency compare to the old one?

A

The new beat frequency will be exactly the same.

Old beat frequency: |220-224| = 4
New beat frequency: |232-228| = 4

58
Q

Define:

refraction

A

Refraction occurs whenever a wave transitions from one medium to another. Typically, the wave will have different properties in the incident medium than in the propagating medium.

A refracting wave can change speed, amplitude, and direction, but its frequency is always conserved.

59
Q

A wave refracts from Medium 1 to Medium 2 at a certain angle to the normal. If the wave moves much slower in Medium 2, will the angle be larger or smaller in that medium than it was in Medium 1?

A

The angle will be smaller (closer to the normal) in Medium 2.

Waves always bend closer to the normal in the medium where they travel more slowly. This occurs because the momentum of the component of the wave parallel to the normal must be conserved. If the wave is moving more slowly, it must move closer to the normal to conserve this quantity.

60
Q

Define:

diffraction

A

Diffraction occurs when a wave is incident on a narrow gap in a barrier. The gap acts as a point source, and the wave on the far side of the gap propagates in circular waveforms.

Diffraction can be thought of as the spreading of a wave.

61
Q

In Young’s double slit experiment, why does light incident on two narrow slits in a barrier form a fringe pattern on a far screen?

A

Young’s results are explained by diffraction. The two slits act as point sources, and the waves interfere with one another depending on the difference between their path lengths. This creates the classic fringe pattern.