b. At x = 0
A mass is oscillating on a frictionless surface at the end of a horizontal spring. Where, if anywhere, is the acceleration of the mass zero.
a. At x = -A
b. At x = 0
c. At x = +A
d. at both x = -A and x = +A
e. nowhere
a. At x = -A
b. At x = 0
c. At x = +A
d. at both x = -A and x = +A
e. nowhere
c. 80 seconds
If an oscillating mass has a frequency of 1.25 z, it makes 100 oscillations in
a. 12.5 seconds
b. 125 seconds
c. 80 seconds
d. 8.0 seconds
a. 12.5 seconds
b. 125 seconds
c. 80 seconds
d. 8.0 seconds
1/2 second
If the frequency is 2 cycles per second, how long is each cycle?
Distance of the mass from the equilibrium point at any moment
Displacement
Maximum displacement
Amplitude
Complete to-and-fro motion from some initial point back to that same point
Cycle
Time to complete one cycle
Period
Number of complete cycles per second
Frequency
1. There must be a position of stable equilibrium
2. There must be a negative restoring force.
3. The resistive forces in the system must be reasonably small.
2. There must be a negative restoring force.
3. The resistive forces in the system must be reasonably small.
Conditions that lead to simple harmonic motion
B and D (they give the force as minus a constant times a displacement. The displacement doesn’t have to be x, but the minus sign is required to restore the system to equilibrium)
Which of the following forces would cause an object to move in simple harmonic motion?
a. F = -0.5x²
b. F = -2.3y
c. F = 8.6x
d. F = -4θ
a. F = -0.5x²
b. F = -2.3y
c. F = 8.6x
d. F = -4θ
a. Since E is proportional to the square of A, stretching it twice as far quadruples the energy
b. When amplitude is doubled, max velocity must be doubled.
c. Since force is twice as great when we stretch the spring twice as far, the acceleration is also twice as great.
b. When amplitude is doubled, max velocity must be doubled.
c. Since force is twice as great when we stretch the spring twice as far, the acceleration is also twice as great.
Suppose a spring is stretched twice as far (to x=2A). What happens to
a. the energy of the system?
b. the max velocity of the oscillating mass?
c. the max acceleration of the mass?
a. the energy of the system?
b. the max velocity of the oscillating mass?
c. the max acceleration of the mass?
a. Increases
b. Increases
c. Increases
b. Increases
c. Increases
Suppose a spring is compressed to x = -A, but is given a push to the right so that the initial speed of the mass is v₀. What effect does this have on
a. the energy of the system?
b. the max velocity of the oscillating mass?
c. the max acceleration of the mass?
a. the energy of the system?
b. the max velocity of the oscillating mass?
c. the max acceleration of the mass?
c. quadrupled
By how much should the mass on the end of a spring be changed to halve the frequency of its oscillations?
a. No change
b. Doubled
c. Quadrupled
d. Halved
e. Quartered
a. No change
b. Doubled
c. Quadrupled
d. Halved
e. Quartered
a and b;
larger mass = longer period
larger k/stiffer spring = shorter period
Which of the following factors affect the period in simple harmonic motion?
a. mass of the object
b. spring constant (stiffness)
c. amplitude
a. mass of the object
b. spring constant (stiffness)
c. amplitude
Mass must be quadrupled
What must happen to the mass to double the period?
c. the same, or very close to it
(amplitude does not affect frequency)
A simple pendulum consists of mass m (the “bob” hanging on the end of a thin string of length l and negligible mass. The bob is pulled sideways so the string makes a 5.0⁰ angle to the vertical. When released, it oscillates back at forth at frequency f. If the pendulum is started at a 10⁰ angle instead, its frequency would be
a. twice as great
b. half as great
c. the same, or very close to it
d. not quite twice as great
e. a bit more than half as great
a. twice as great
b. half as great
c. the same, or very close to it
d. not quite twice as great
e. a bit more than half as great
a. slightly slower
(gravity decreases so period increases — takes longer to complete one cycle)
If a simple pendulum is taken from sea level to the top of a high mountain and started at the same angle of 5⁰, it would oscillate at the top of the mountain
a. slightly slower
b. slightly faster
c. at exactly the same frequency
d. not at all – it would stop
e. none of these
a. slightly slower
b. slightly faster
c. at exactly the same frequency
d. not at all – it would stop
e. none of these
Amplitude of any real oscillating spring or swinging pendulum slowly decreases in time until oscillations stop altogether; generally due to air resistance and internal friction within the oscillating system. The energy is dissipated to thermal energy
Damped harmonic motion
SHM is easier to deal w/ mathematically. If damping isn’t large, the oscillations can be thought of as SHM on which the damping is superimposed.
Since natural oscillating systems are damped in general, why do we talk about undamped SHM?
1. Underdamped – system makes several oscillations before coming to rest.
2. Overdamped – damping is so large that there is no oscillation and the system takes a long time to come to rest
3. Critical damping – displacement reaches 0 quickly.
2. Overdamped – damping is so large that there is no oscillation and the system takes a long time to come to rest
3. Critical damping – displacement reaches 0 quickly.
3 Common Cases of Heavily Damped Systems
door closing mechanisms and shock absorbers in a car — designed to give critical damping
Large buildings in CA are retrofitted w/ huge dampers to reduce earthquake damage. They reduce the amplitude and the acceleration of movement when it hits
Practical damped systems; What type of damping do they give?
Underdamping occurs: the door of a room slams and a car bounces up and down several times when it hits a bump
What happens when practical damping systems wear out?
When an oscillating system is set into motion, it oscillates at its natural frequency without any external forces acting on it
Natural frequency/Resonant Frequency
System may have an external force applied to it that has its own particular frequency.
Forced oscillations
Effect of increased amplitude when f = f₀
Resonance
Propagate as oscillations of matter; include water waves and waves on a rope or cord
Mechanical Waves
– Can move large distances, but the medium (i.e. water, rope) has limited movement, oscillating about an equilibrium point
– Wave itself is not matter, but the wave pattern can travel IN matter
– A wave consists of oscillations that move w/o carrying matter with them
-They carry energy from one place to another
– Wave itself is not matter, but the wave pattern can travel IN matter
– A wave consists of oscillations that move w/o carrying matter with them
-They carry energy from one place to another
Properties of Waves
No. Velocities are different with regards to magnitude and direction. The wave on the rope moves along the tabletop, but each piece only vibrates to and fro, perpendicular to the traveling wave.
Is the velocity of a wave moving along a rope the same as the velocity of a particle of the rope?
c. The waves only make the water move up and down, but the waves do carry energy outward, away from where the rock hit.
You drop a rock into a pond, and water wave spread out in circles.
a. The waves carry outward, away from where the rock hit. That moving water carries energy outward.
b. The waves only make the water move up and down. No energy is carried outward from where the rock hit.
c. The waves only make the water move up and down, but the waves do carry energy outward, away from where the rock hit.
a. The waves carry outward, away from where the rock hit. That moving water carries energy outward.
b. The waves only make the water move up and down. No energy is carried outward from where the rock hit.
c. The waves only make the water move up and down, but the waves do carry energy outward, away from where the rock hit.
Single wave bump. Can be formed on a cord by a quick up-and-down motion of the hand; Source of traveling wave pulse is a disturbance (or vibration), and cohesive forces between adjacent sections of cord cause the pulse to travel.
Pulse wave
Source is a disturbance that is continuous and oscillating.
Continuous/Periodic Wave
Vibration; it is the vibration that propagates outward and constitutes the wave
What is the source of any wave?
high points on a wave
Crests
low points on a wave
Troughs
max height of a crest or depth of a trough. The total swing from a crest to a trough is 2A
Amplitude (Wave)
distance between two successive crests; also equal to the distance between any two successive identical points on the way
Wavelength
number of crests/complete cycles that pass a given point per unit time
Frequency (Wave)
1/f; time elapsed between two successive crests passing the same point in space
Period (Wave)
d. The period is 0.5 s
You notice a water wave pass by the end of a pier, with about 0.5 seconds between crests. Therefore
a. The frequency is 0.5 Hz
b. The velocity is 0.5 m/s
c. The wavelength is 0.5 m
d. The period is 0.5 s
a. The frequency is 0.5 Hz
b. The velocity is 0.5 m/s
c. The wavelength is 0.5 m
d. The period is 0.5 s
Wave in which particles of the medium move perpendicular to the direction that the wave move
Transverse Wave
A wave in which the oscillations are parallel to the direction of wave propagation; formed on a stretched spring or Slinky — important example is a sound wave in air
Characterized with compressions and expansions (equivalent to crests and troughs)
Longitudinal Wave
Transverse S (shear) waves and Longitudinal P (pressure/ compression) waves. L & T waves can travel through a solid, but only L waves can propagate through fluid
–> Earth’s core must be liquid because after an earthquake L waves are detected across Earth.
Earthquakes
travel along the boundary between two materials (wave on water is actually a surface wave that moves on the boundary between water and air)
Surface Waves
occurs when two waves pass through the same region of space at the same time
Interference
In the region where two waves overlap, the resultant displacement is the algebraic sum of their separate displacements
Principle of Superposition
If two waves have opposite displacements at the instant they pass one another, they add to zero
Destructive Interference
At the instant two pulses overlap, they produce a resultant displacement that is greater than the displacement of either separate pulse
Constructive Interference
Wave that does not appear to be traveling. The cord appears to have segments that oscillate up and down in a fixed pattern
Standing Wave
Points of destructive interference, where the cord remains still at all times
Nodes
Points of constructive interference, where the cord oscillates with maximum amplitudes
Antinodes
Frequencies at which standing waves are produced
Natural Frequencies/Resonant Frequencies
has 1 antinode; also known as first harmonic
Fundamental Frequency
d. move to the left until it reaches x=-A and then begin to move to the right
After the block is released from x = A, it will
a. remain at rest
b. move to the left until it reaches equilibrium and stops there
c. move to the left until it reaches x=-A and stop there
d. move to the left until it reaches x=-A and then begin to move to the right
a. remain at rest
b. move to the left until it reaches equilibrium and stops there
c. move to the left until it reaches x=-A and stop there
d. move to the left until it reaches x=-A and then begin to move to the right
c. halved
If the period is doubled, the frequency is
a. unchanged
b. doubled
c. halved
a. unchanged
b. doubled
c. halved
f = 10 Hz
An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequency f?
T = 0.025 s
If the frequency is 40 Hz, what is the period T ?
b. It decreases the amplitude
What effect does dropping the sandbag out of the cart at the equilibrium position have on the amplitude of your oscillation?
a. It increases the amplitude
b. It decreases the amplitude
c. It has no effect on the amplitude
a. It increases the amplitude
b. It decreases the amplitude
c. It has no effect on the amplitude
c. The wavelength doubles but the wave speed is unchanged.
If the period of the oscillator doubles, what happens to the wavelength and wave speed?
a. The wavelength is unchanged but the wave speed doubles.
b. The wavelength is halved but the wave speed is unchanged.
c. The wavelength doubles but the wave speed is unchanged.
a. The wavelength is unchanged but the wave speed doubles.
b. The wavelength is halved but the wave speed is unchanged.
c. The wavelength doubles but the wave speed is unchanged.
c. Both wavelength and wave speed are unchanged.
The wavelength can be varied only by changing the frequency, or alternatively the period, of the oscillator that creates the waves.
If the amplitude of the oscillator doubles, what happens to the wavelength and wave speed?
a. The wavelength doubles but the wave speed is unchanged.
b. The wavelength is unchanged but the wave speed doubles.
c. Both wavelength and wave speed are unchanged.
a. The wavelength doubles but the wave speed is unchanged.
b. The wavelength is unchanged but the wave speed doubles.
c. Both wavelength and wave speed are unchanged.
b and c; wave speed and wavelength
If the strings have different thicknesses, which of the following parameters, if any, will be different in the two strings?
a. wave frequency
b. wave speed
c. wavelength
d. none of the above
a. wave frequency
b. wave speed
c. wavelength
d. none of the above
d. none of the above
If the strings have the same thickness but different lengths, which of the following parameters, if any, will be different in the two strings?
a. wave frequency
b. wave speed
c. wavelength
d. none of the above
a. wave frequency
b. wave speed
c. wavelength
d. none of the above
When a wave travels along a string, the wave speed depends exclusively on the properties of the string, whereas the wave frequency is set by the oscillator that creates the waves. The wavelength is a quantity that can vary if either the wave speed or the wave frequency is changed. Thus, it can be modified by changing either the motion of the oscillator or the properties of the string.
Summary of Wave Parameters
b. 2f₀
An open organ pipe (i.e., a pipe open at both ends) of length L0 has a fundamental frequency f₀. If the organ pipe is cut in half, what is the new fundamental frequency?
a. 4f₀
b. 2f₀
c. f₀
d. f₀/2
e. f₀/4
a. 4f₀
b. 2f₀
c. f₀
d. f₀/2
e. f₀/4
c. f₀
The fundamental frequency of a half-length closed pipe is equal to that of a full-length open pipe.
After being cut in half in Part A, the organ pipe is closed off at one end. What is the new fundamental frequency?
a. 4f₀
b. 2f₀
c. f₀
d. f₀/2
e. f₀/4
a. 4f₀
b. 2f₀
c. f₀
d. f₀/2
e. f₀/4
a. 3f₀
The air from the pipe in Part B (i.e., the original pipe after being cut in half and closed off at one end) is replaced with helium. (The speed of sound in helium is about three times faster than in air.). What is the approximate new fundamental frequency?
a. 3f₀
b. 2f₀
c. f₀
d. f₀/2
e. f₀/3
a. 3f₀
b. 2f₀
c. f₀
d. f₀/2
e. f₀/3
b, c, d
An object oscillates back and forth on the end of a spring. Which of the following statements are true at some time during the course of the motion? Check all that apply.
a. The object can have zero velocity and, simultaneously, zero acceleration.
b. The object can have nonzero velocity and nonzero acceleration simultaneously.
c. The object can have zero velocity and, simultaneously, nonzero acceleration.
d. The object can have zero acceleration and, simultaneously, nonzero velocity
a. The object can have zero velocity and, simultaneously, zero acceleration.
b. The object can have nonzero velocity and nonzero acceleration simultaneously.
c. The object can have zero velocity and, simultaneously, nonzero acceleration.
d. The object can have zero acceleration and, simultaneously, nonzero velocity
b. 4M
An object of mass M oscillates on the end of a spring. To double the period, replace the object with one of mass
a. 2M
b. 4M
c. M/2
d. M/4
e. None of the above
a. 2M
b. 4M
c. M/2
d. M/4
e. None of the above
b. decreasing m
11-3: smaller mass = smaller period = larger frequency
An object of mass m rests on a frictionless surface and is attached to a horizontal ideal spring with spring constant k. The system oscillates with amplitude A. The oscillation frequency of this system can be increased by
a. increasing A
b. decreasing m
c. decreasing k
d. More than one of the above
e. None of the above
a. increasing A
b. decreasing m
c. decreasing k
d. More than one of the above
e. None of the above
a. Both children swing w/ the same period
At a playground, two young children are on identical swings. One child appears to be about twice as heavy as the other. If you pull them back together the same distance and release them to start them swinging, what will you notice about the oscillations of the two children?
a. Both children swing w/ the same period
b. The lighter child swings with a larger period
c. The heavier child swings with a larger period
a. Both children swing w/ the same period
b. The lighter child swings with a larger period
c. The heavier child swings with a larger period
b. shortening the string
A grandfather clock is “losing” time because its pendulum moves too slowly. Assume that the pendulum is a massive bob at the end of a string. The motion of this pendulum can be sped up by:
a. increasing the mass of the bob
b. shortening the string
c. decreasing the mass of the bob
d. lengthening the string
a. increasing the mass of the bob
b. shortening the string
c. decreasing the mass of the bob
d. lengthening the string
a. Stretching the elastic cord further
use speed of transverse wave equation
Which of the following increases the speed of waves in a stretched elastic cord? (More than one answer may apply.)
a. Stretching the elastic cord further
b. Increasing wave amplitude
c. Increasing the wavelength
d. Increasing the wave frequency
a. Stretching the elastic cord further
b. Increasing wave amplitude
c. Increasing the wavelength
d. Increasing the wave frequency
b. energy but not matter
A wave transports
a. both energy and matter
b. energy but not matter
c. matter but not energy
a. both energy and matter
b. energy but not matter
c. matter but not energy
