# Periodic Motion, Waves & Sound Flashcards

## From wave frequency to the Doppler effect, use these cards to master the topic of Sound, Waves, and Periodic Motion as tested in most introductory undergrad physics courses and even on the AP Physics exam.

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

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

Don’t fall into the trap of thinking the amplitude is the full range from minimum to maximum; that is actually twice the amplitude.

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

The amplitude of the wave is 3 cm.

Don’t make the mistake of thinking the amplitude covers the entire wave, from minimum to maximum; that value (6 cm in this case) is twice the actual amplitude.

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

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

Don’t make the mistake of measuring from one zero of the waveform to the next zero; that value only captures one-half of the oscillation, and represents one-half of the period.

Period is typically represented by T on the AP Physics exam.

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

The period of the wave is 6.5 s.

Don’t make the mistake of thinking the period is measured from one midpoint to the next; that value (3.25 s in this case) is half the period.

Which waveform is oscillating more rapidly, if the period of waveform 1 is twice that of waveform 2?

Waveform 2 is oscillating more rapidly.

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

The frequency of a waveform is calculated as **f = 1/T**, where f is the waveform’s frequency, and T is the waveform’s period.

The common units of frequency on the AP Physics exam are Hz, where 1 Hz is 1 s^{-1}.

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 radian/s.

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 4.5 s, and the frequency f = 1/T, or 1/4.5 s^{-1}.

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

Waveform 2 is oscillating more rapidly.

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

Define:

the **phase** of a waveform

The **phase** of a waveform is the offset of the waveform relative to its origin (zero value).

Although phase can hold any value, on the AP Physics exam the only commonly-tested values are integer amounts of quarter-wavelengths.

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.

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

The two waveforms below differ by one-half of a wave. What is the calculated phase difference in:

- degrees?
- radians?

One-half of a wave corresponds to a phase difference of:

- 180º
- π radians

The two waveforms below differ by one-quarter of a wave. What is the calculated phase difference in:

- degrees?
- radians?

One-quarter of a wave corresponds to a phase difference of:

- 90º
- π/2 radians

Define:

**Hooke’s Law** for spring force

**Hooke’s Law** relates the extension of a spring to the force exerted by the spring.

F_{x} = -kx

Where:

F_{x} = the force exerted by the spring in N

k = the force constant of the spring in N/m

x = the extension of the spring from equilibrium in m

By how much does a vertically-hung spring stretch if its force constant is 1,000 N/m and 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, 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 that results when an object at equlibrium is displaced, and feels a restorative force that is proportional to the displacement.

The most common examples of simple harmonic motion seen on the AP Physics exam are the mass on a spring and the motion of a pendulum.

What is the shape of the graph for the motion of any simple harmonic motion system? (assume no frictional forces)

All simple harmonic motion systems have a sinusoidal graph.

Ex: In the below graph, displacement is graphed vs. time for a mass on a spring system.

How would the graph for a mass on a spring look different if the mass begins to oscillate more quickly?

The peaks of the graph will move closer together.

Ex: In the graph below, the red line represents a system which is oscillating more rapidly.

When is the point of maximum kinetic energy for a system oscillating with simple harmonic motion?

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 below. The slope maximizes as the line crosses through equilibrium, and is at a minimum (0) at the extreme positions.

When is the point of maximum potential energy for a system oscillating with simple harmonic motion?

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 will occur when the kinetic energy is at a minimum.

What is the formula for the angular frequency of an object oscillating on a spring?

For an object oscillating on a spring, the angular frequency is:

ω = (k/m)^{½}

Where:

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

m = the mass of the object

Two objects are oscillating on identical springs. Object 1 has a mass of 5 kg, while the mass of object 2 is 10 kg. Which one oscillates more rapidly?

Object 1 oscillates at the higher frequency.

Remember: the frequency of an object oscillating on a spring is (k/m)^{½}. The frequency is inversely proportional to the square root of the mass, so the larger the mass, the lower the frequency.

Two identical objects are oscillating on springs. Spring 1 has a force constant of 5,000 N/m, while the force constant of spring 2 is 10,000 N/m. Which one oscillates more rapidly?

The mass on spring 2 oscillates at the higher frequency.

Remember: the frequency of an object oscillating on a spring is (k/m)^{½}. The frequency is directly proportional to the square root of the force constant, so the larger the force constant, the higher the frequency.

Two identical objects are suspended from identical springs. One is displaced 10 cm from equilibrium, the other is displaced 20 cm from equilibrium, and both are allowed to oscillate. Which one oscillates more rapidly?

The two springs oscillate at the same frequency.

Remember: 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 amplitude of the oscillation.

What is the potential energy of a spring displaced from equilibrium?

The potential energy of any spring stretched a distance x from equlibrium is:

U = ½kx^{2}

Where k is the force constant of the spring.

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

The maximum value of the potential energy for a mass oscillating on a spring is U_{max} = ½kA^{2}.

Remember: the potential energy for the system for any value x of the spring’s extension is ½kx^{2}. The maximum value of x is A, the amplitude. Plugging in A for x gives the maximum value of U.

What is the kinetic energy of a mass oscillating on a spring?

The kinetic energy of a mass oscillating on a spring is:

KE = ½mv^{2}

Where:

m = the mass of the object oscillating

v = the velocity of the object at that moment

What is the maximum value of the kinetic energy of a mass oscillating on a spring?

KE_{max} = ½kA^{2}.

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 full extension. At the equilibrium position, all the energy in the system is in the form of kinetic energy.

What is the maximum value of the velocity of a mass oscillating on a spring?

v_{max} = A(k/m)^{½} = Aω.

The maximum value of the kinetic energy is ½kA^{2}. Setting this equal to ½mv^{2} and solving for v yields the final answer.

What is the angular frequency of an oscillating pendulum?

f = (g/L)^{½}

Where:

f = the pendulum oscillation frequency in Hz

g = gravitational acceleration in m/s^{2}

L = length of the string of the pendulum

What is the difference in the oscillation frequency of two different pendulums, if pendulum 1 has a string four times the length of pendulum 2?

Pendulum 2 has a frequency twice as high as the frequency 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 the string’s length. Since pendulum 1 has a string 4 times as long, its frequency is 1/√4 that of pendulum 2, or half as much.

Define:

**Transverse wave**

In a **transverse wave**, the particles oscillating move with displacement perpendicular to the direction of propagation.

What are some classic examples of transverse waves?

Transverse waves include:

- light waves (electromagnetic waves)
- string waves
- pond waves*
- stadium waves

*Technically water waves also fall under the classification of “surface waves”, but the AP Physics exam does not require that definition.

Define:

**Longitudinal wave**

In a **longitudinal wave**, the particles oscillating move with displacement parallel to the direction of propagation.

What are some classic examples of longitudinal waves?

Logitudinal waves include:

- sound waves
- stretched slinky waves (spring waves)

Calculate the period of the wave shown below:

The period is 5 seconds.

A full wave cycle is any 360 degree segment and is usually easiest to see as peak-to-peak, or valley-to-valley. Notice that the wave valley hits exactly at 0, 5, 10, etc.

Calculate the frequency of the wave below:

The frequency is .2 Hz.

Since frequency = 1/T, and the period is 5 seconds, this gives f = 1/5 = .2

Calculate the amplitude of the wave below:

The amplitude is 17 meters.

Amplitude is the greatest positive displacement from zero of a wave, measured at its peak.