## Unit 5: Oscillatory Motion and Mechanical Waves Summary

In Unit we learned about Oscillatory Motion and Mechanical waves where we learned horizontal and vertical spring systems that demonstrated Hooke's law. After that we bounced into Mechanical waves where we learned about super position and interference waves, what waves are and the Doppler effect.

Interference

**Terminology:****Period**- is the time required to complete one cycle**Frequency**- is the number of cycles per second**Oscillation**- is a motion that repeats itself in a regular cycle**Cycle**- is motion in which the period of each cycle is constant**Hooke's law**- The deformation of an object is proportional to the force causing the deformation**Restoring force**- is a force acting opposite to the displacement to move the object back to its equilibrium position**Simple harmonic motion**- is oscillatory motion where the resting force is proportional to the displacement of the mass.**Simple harmonic oscillator**- a mass on the end of a spring**Forced frequency**- is the number of vibrations executed by a body on being disturbed from its mean position by application of external force in vacuum or space**Resonant frequency**- the natural frequency of vibration of an object**Mechanical resonance**- the increase in amplitude of oscillation of a system as a result of a periodic force hows frequency is equal or very close to the resonant**Medium**- s simply the material through which the disturbance is moving**Wave**- can be described as a disturbance that travels through a medium**Amplitude**- is the magnitude of greatest displacement from the equilibrium**Crest**- is the high point or maximum of a wave length**Trough**- is the low point or minimum of a wave length**Wavelength**- is a complete cycle of a crest followed by a trough**In phase waves**- 2 crests or 2 troughs overlap, produces a constructive interference**Out of phase waves**- crest and trough overlap each other, produces a destructive interference**Pulse**- is a short disturbance, usually just a crest or trough**Longitudinal wave**- the displacement of the medium is parallel to the direction of the wave.**Transverse wave**- the displacement of the medium is perpendicular to the direction of the wave travel**Ray**- is a line that indicates only the direction of the wave front at any point where the ray and the wave front intersectInterference

**Node**- are located every half wave length from the closed end, they are located at the closed end.**Antinode**- are located one quarter wave length from the closed end and then every half - wavelength from that point.**Standing wave**- is a combination of two waves moving in opposite directions, each having the same amplitude and frequency.**Doppler effec**t - can be described as the affect of moving a source of waves## Simple Harmonic Motion Horizontal Mass Systems

Simple harmonic motion is simply oscillatory motion. Here is a video that in-depth explains it. Video

## Simple Harmonic Motion Vertical Mass Systems

## Position, Velocity, Acceleration, and Time Relationships

Simple Harmonic Motion presents us with 4 main variables, Acceleration, Time, Velocity and Position, all of which can be calculated. Video exemplifying a few ways to calculate period, frequency and amplitude.

## The properties of Waves

In this sub unit we learned about properties of waves, an example is water transporting the medium of the wave and the undisturbed surface of the water is known as the equilibrium position. Regions where the water rises above the equilibrium position are called crests and regions where the water is lower than the equilibrium position are called troughs. In the crest or trough, the magnitude of greatest displacement from the equilibrium is defined as the waves amplitude. A complete cycle of a crest followed by a trough is called a wave length. There is also a relationship within this. The relationship between waves and energy is that energy produce waves. An example of energy producing waves is a speaker with sound carrying energy therefore producing sound waves. Without energy, waves wouldn't be able to produce.

## Transverse and Longitudinal Waves

A longitudinal wave is a wave in which particles of the medium move in a direction parallel to the direction that the wave moves.

A Transverse wave is a wave in which particles of the medium move in a direction perpendicular to the direction that the wave moves.

Video

A Transverse wave is a wave in which particles of the medium move in a direction perpendicular to the direction that the wave moves.

Video

## Super position and Interference

Wave interference occurs when two waves meet while traveling along the same medium disturbing each other.

Superposition occurs when waves pass through each other without being disturbed.

Constructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the same direction.

Destructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction.

Video of wave interference video

An example of super position video

Constructive vs Destructive video

In phase waves - 2 crests or 2 troughs overlap, produces a constructive interference

out of phase waves - crest and trough overlap each other, produces a destructive interference

Superposition occurs when waves pass through each other without being disturbed.

Constructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the same direction.

Destructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction.

Video of wave interference video

An example of super position video

Constructive vs Destructive video

In phase waves - 2 crests or 2 troughs overlap, produces a constructive interference

out of phase waves - crest and trough overlap each other, produces a destructive interference

## Doppler Effect

Have you ever stood outside by a road watching cars pass by and you can hear different pitches of sound when the car is coming towards you and away, this is called the Doppler effect.

Video on the Doppler Effect

Video on the Doppler Effect

## Formulas

## Oscillatory Motion and Mechanical Wave examples

1. Earthquakes waves that travel along Earth's surface can have periods of up to 5.00 minutes. What is their frequency?

T = 1 / f

f = 1 / T

f = 1 / 300s

f = 3.33 x 10 ^ -3 Hz

2. A hummingbird can hover when it flaps its wings with a frequency of 78 Hz. What is the period of the wing's motion?

T = 1 / f

T = 1 / 78 Hz

T = .013 s

3. A dog happy to see its owner wags its tail 2.50 times a second. What is the period of the wagging tail? How many wags of its tail will the dog make in 1 minute?

T = 1 / f

T = 1 / 2.5 Hz

T = .4s

1 x 60s = 60s.

T x f = wags

60s x 2.5Hz = 150 wags in 1 minute

4. A spring requires a force of 100.0 N to compress it a displacement of 4.0 cm. What is its spring constant?

F = -kx

k = F / x

k = 100.0 N / .04m

k = 2.5 x 10 ^ 3 N/m

5. Give people with a combined mass of 275.0 kg get into a car. The car's four springs are each compressed a distance of 5.00 cm. Determine the spring constant of the springs. Assume the mass is distributed evenly.

F = -kx

k = F / x

k = 275.0 kg x 9.81 m/s ^ 2 / .05m

k = 53955 / 4 (there are 4 springs)

k = 1.35 x 10 ^ 4 N/m

6. Determine the restoring force of a pendulum that is pulled an angle of 12.0 degree left of the vertical. The mass of the bob is 300.0 g

F = m g sin ( delta )

F = .300 kg x 9.81 m/s ^ 2 x sin ( 12 )

F = .612 N

7. A 50.0 g mass is attached to a spring with a constant of 4.00 N/m. The mass oscillates with an amplitude of 1.12m What is its maximum speed?

v max = A square root of k/m

v max = 1.12m square root 4.00 N/m / .05 kg

v max = 10 m/s

8. An instructor sets up an oscillating vertical mass-system k 6.05 N/m. The maximum displacement is 81.7 cm and the maximum speed is 2.05 m/s. What is the mass of the oscillator?

v max = A square root k / m

v ^ 2 max = A ^ 2 k / m

m = A ^ 2 k / v ^ 2

m = .817m ^ 2 x 6.05 N/m / 2.05 m/s ^ 2

m = .961 kg

9. What is the gravitational field strength on Mercury if a .500m pendulum swings with a period of 2.30 s?

T = 2pi square root of l / g

g = 4 pi ^ 2 x l / T ^ 2

g = 4 pi ^ 2 x .500m / 2.30s ^ 2

g = 3.73 N/ kg down

10. What period would a 30 cm pendulum have on mars, where the gravitational field strength is 3.71 N/kg down?

T = 2pi square root of l / g

T = 2pi square root of .30m / 3.71 N/kg down

T = 1.79 s

11. A pulse is generated in a spring where it travels 5.30 m/s. If the time to generate the pulse is .640s, what will be its length?

l = v x t

l = 5.30 m/s x .640 s

l = 3.39 m

12. pulse moves along a spring at a speed of 3.60 m/s. If the length of the pulse is 2.50m, how long did it take to generate the pulse?

l = v x t

t = l / v

t = .694 s

13. A tuning fork with a frequency of 256 Hz is held above a closed air column while the column is gradually increased in length. At what lengths for this air column would the first 4 resonant points be found, if the speed of sound is 330 m/s

v = f lambda

lambda = v / f

lambda = 330 m/s / 256 Hz

lambda = 1.29

1.29 x (1/4) = .322m

From now every third will be heard.

.322m x 3 = .967m

.322m x 5 = 1.61 m

.322m x 7 = 2.26m

14. You are crossing in a crosswalk when an approaching driver blows his horn. If the true frequency of the horn is 264 Hz and the car is approaching you at a speed of 60.0 km/h, what is the apparent frequency of the horn? Assume the speed of sound in air is 340 m/s.

fd = ( Vw / Vw + or - Vs) fs

fd = ( 340 m/s / 340 m/s - 16.67 m/s) x 264 Hz

fd = 278 Hz

Answer:Answer:

T = 1 / f

f = 1 / T

f = 1 / 300s

f = 3.33 x 10 ^ -3 Hz

2. A hummingbird can hover when it flaps its wings with a frequency of 78 Hz. What is the period of the wing's motion?

**Answer:**T = 1 / f

T = 1 / 78 Hz

T = .013 s

3. A dog happy to see its owner wags its tail 2.50 times a second. What is the period of the wagging tail? How many wags of its tail will the dog make in 1 minute?

**Answer:**T = 1 / f

T = 1 / 2.5 Hz

T = .4s

1 x 60s = 60s.

T x f = wags

60s x 2.5Hz = 150 wags in 1 minute

4. A spring requires a force of 100.0 N to compress it a displacement of 4.0 cm. What is its spring constant?

**Answer:**F = -kx

k = F / x

k = 100.0 N / .04m

k = 2.5 x 10 ^ 3 N/m

5. Give people with a combined mass of 275.0 kg get into a car. The car's four springs are each compressed a distance of 5.00 cm. Determine the spring constant of the springs. Assume the mass is distributed evenly.

**Answer:**F = -kx

k = F / x

k = 275.0 kg x 9.81 m/s ^ 2 / .05m

k = 53955 / 4 (there are 4 springs)

k = 1.35 x 10 ^ 4 N/m

6. Determine the restoring force of a pendulum that is pulled an angle of 12.0 degree left of the vertical. The mass of the bob is 300.0 g

**Answer:**F = m g sin ( delta )

F = .300 kg x 9.81 m/s ^ 2 x sin ( 12 )

F = .612 N

7. A 50.0 g mass is attached to a spring with a constant of 4.00 N/m. The mass oscillates with an amplitude of 1.12m What is its maximum speed?

**Answer:**v max = A square root of k/m

v max = 1.12m square root 4.00 N/m / .05 kg

v max = 10 m/s

8. An instructor sets up an oscillating vertical mass-system k 6.05 N/m. The maximum displacement is 81.7 cm and the maximum speed is 2.05 m/s. What is the mass of the oscillator?

**Answer:**v max = A square root k / m

v ^ 2 max = A ^ 2 k / m

m = A ^ 2 k / v ^ 2

m = .817m ^ 2 x 6.05 N/m / 2.05 m/s ^ 2

m = .961 kg

9. What is the gravitational field strength on Mercury if a .500m pendulum swings with a period of 2.30 s?

**Answer:**T = 2pi square root of l / g

g = 4 pi ^ 2 x l / T ^ 2

g = 4 pi ^ 2 x .500m / 2.30s ^ 2

g = 3.73 N/ kg down

10. What period would a 30 cm pendulum have on mars, where the gravitational field strength is 3.71 N/kg down?

**Answer:**T = 2pi square root of l / g

T = 2pi square root of .30m / 3.71 N/kg down

T = 1.79 s

11. A pulse is generated in a spring where it travels 5.30 m/s. If the time to generate the pulse is .640s, what will be its length?

Answer:Answer:

l = v x t

l = 5.30 m/s x .640 s

l = 3.39 m

12. pulse moves along a spring at a speed of 3.60 m/s. If the length of the pulse is 2.50m, how long did it take to generate the pulse?

**Answer:**l = v x t

t = l / v

t = .694 s

13. A tuning fork with a frequency of 256 Hz is held above a closed air column while the column is gradually increased in length. At what lengths for this air column would the first 4 resonant points be found, if the speed of sound is 330 m/s

Answer:Answer:

v = f lambda

lambda = v / f

lambda = 330 m/s / 256 Hz

lambda = 1.29

1.29 x (1/4) = .322m

From now every third will be heard.

.322m x 3 = .967m

.322m x 5 = 1.61 m

.322m x 7 = 2.26m

14. You are crossing in a crosswalk when an approaching driver blows his horn. If the true frequency of the horn is 264 Hz and the car is approaching you at a speed of 60.0 km/h, what is the apparent frequency of the horn? Assume the speed of sound in air is 340 m/s.

Answer:Answer:

fd = ( Vw / Vw + or - Vs) fs

fd = ( 340 m/s / 340 m/s - 16.67 m/s) x 264 Hz

fd = 278 Hz