Periodic Motion & Waves

Question 1

Where is the amplitude of the waveform shown below?

Answer 1

It 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.

Question 2

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

Answer 2

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.

Question 3

What is the period of the waveform shown below?

Answer 3

It 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.

Question 4

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

Answer 4

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.

Question 5

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

Answer 5

Waveform 2

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.

Question 6

What is the frequency of a waveform?

Answer 6

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.

Question 7

What units are used to measure frequency?

Answer 7

hertz (Hz)

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

Question 8

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

Answer 8

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.

Question 9

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

Answer 9

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.

Question 10

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

Answer 10

0.2 Hz

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

Question 11

Which waveform is oscillating more rapidly, if the frequency of waveform 1 is 100 Hz, while that of waveform 2 is 200 Hz?

Answer 11

Waveform 2

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.

Question 12

Define:

phase

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

Answer 12

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.

Question 13

What is the phase difference between two waveforms?

Answer 13

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.

Question 14

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

Answer 14

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.

Question 15

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

Answer 15

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.

Question 16

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

Answer 16

Hooke’s Law or F_x = -kx

In this equation,

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

Question 17

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?

Answer 17

10 cm

According to Hooke’s Law, F_x = -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

Question 18

Define:

simple harmonic motion

(SHM)

Answer 18

It is the motion produced when an object at equilibrium is displaced and feels a restorative force proportional to the displacement.

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

Question 19

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

Assume that no frictional forces are present.

Answer 19

All simple harmonic motion systems have sinusoidal graphs.

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

Question 20

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

Answer 20

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.

Question 21

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

Answer 21

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.

Question 22

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

Answer 22

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.

Question 23

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

Answer 23

ω = (k/m)^½

Where:

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

Question 24

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?

Answer 24

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.

Question 25

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?

Answer 25

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.

Question 26

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?

Answer 26

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.

Question 27

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

Answer 27

U = ½kx²

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

Question 28

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

Answer 28

U_max = 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.

Question 29

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

Answer 29

KE = ½mv²

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

Question 30

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

Answer 30

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.

Question 31

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

Answer 31

v_max = 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.

Question 32

What formula gives the frequency of an oscillating pendulum?

Answer 32

f = (g/L)^½

Where:

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

Question 33

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

Answer 33

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.

Question 34

How does a transverse wave propagate through a medium?

Answer 34

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

Question 35

What are some classic examples of transverse waves?

Answer 35

• 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.

Question 36

How does a longitudinal wave propagate through a medium?

Answer 36

The oscillating particles are displaced parallel to the direction of wave propagation.

Question 37

What are some classic examples of longitudinal waves?

Answer 37

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

Question 38

Calculate the period of the wave shown below:

Answer 38

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.

Question 39

Calculate the frequency of the wave shown below:

Answer 39

0.2 Hz

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

Question 40

Calculate the amplitude of the wave shown below:

Answer 40

17 meters

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

Question 41

Calculate the wavelength of the wave shown below:

Answer 41

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.

Question 42

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

Answer 42

1 m/s

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

Question 43

Define:

intensity

Answer 43

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.

Question 44

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

Answer 44

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

I α A²

Question 45

When might constructive interference be observed?

Answer 45

This 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.

Question 46

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

Answer 46

constructive interference

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.

Question 47

When might destructive interference be observed?

Answer 47

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.

Question 48

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

Answer 48

destructive interference

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.

Question 49

Define:

resonance

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

Answer 49

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.

Question 50

Define:

standing wave

Answer 50

It 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.

Question 51

Define:

node

Answer 51

It 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.

Question 52

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

Answer 52

It 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.

Question 53

Define:

antinode

Answer 53

It 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.

Question 54

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

Answer 54

It 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.

Question 55

Define:

beat frequency

Answer 55

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.

Question 56

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

Answer 56

The frequency of the beats is given by the absolute value of their difference:

f_beat = |f₁ - f₂|.

Question 57

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?

Answer 57

It will be exactly the same.

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

Question 58

Define:

refraction

Answer 58

It 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.

Question 59

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?

Answer 59

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.

Question 60

Define:

diffraction

Answer 60

It 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.

Question 61

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?

Answer 61

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.