# Periodic Motion & Waves Flashcards

## Waves and periodic motion have countless applications to both the Chemical & Physical Foundations section of the MCAT and to real life. Use these cards to master frequency, simple harmonic motion, standing waves, and more, exactly as they appear on the exam.

Where is the **amplitude** of the waveform shown below?

**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.

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

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.

What is the **period** of the waveform shown below?

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.

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

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.

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

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.

What is the **frequency** of a waveform?

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.

What units are used to measure frequency?

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.

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

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.

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

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.

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

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}.

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

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.

Define:

**phase**

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

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.

What is the **phase difference** between two waveforms?

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.

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

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.

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

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.

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

**Hooke’s Law**, or **F _{x} = -kx**, relates force and displacement for a spring.

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)

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?

The spring stretches by 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

Define:

**simple harmonic motion (SHM)**

**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.

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

Assume that no frictional forces are present.

All simple harmonic motion systems have sinusoidal graphs.

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

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

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.

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

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.

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

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.

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

ω = (k/m)^{½}

Where:

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

m = mass of the object (kg)

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?

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.