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Level 108

Simple Harmonic Motion


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Hookes Law
the amount of stretch or compression of an elastic material is directly proportional to the applied force
medium
A substance through which signals can travel (e.g. air for sound waves).
Mechanical wave
A wave that requires a medium through which to travel
Nodes
The points at which two waves cancel in a standing wave.
Antinodes
The positions on a standing wave where the largest amplitudes occur.
What is (mechanical) oscillation?
Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. The term 'vibration' is precisely used to desc…
What is simple harmonic motion?
Oscillatory motion under a retarding force proportional to the amount of displacement from an equilibrium position. An object exhibits SHM only if the force F on it towards a point (the equilibrium position) is proporti…
What is a restoring force?
The restoring force is a function only of position of the mass or particle. It is always directed back toward the equilibrium position of the system. The restoring force is often referred to in si…
No, because the acceleration is not constant.
Do the usual equations of motion apply for SHM? Why?
What is the amplitude of an object in SHM?
The amplitude A is the maximum distance it reaches from the equilibrium position.
What is the period of an object in SHM?
The period T is the time for one oscillation (e.g. from y = A to y = -A and back to y = A).
What is the frequency of an object in SHM?
The frequency f of the motion is the number of oscillations in 1 s. (f = 1/T and T = 1/f)
What is the angular frequency of an object in SHM?
The angular frequency ω is given by ω = 2πf. This relates to the reference circle and circular motion.
maximum displacement
a
maximum velocity
vmax = Aω
maximum acceleration
amax = Aω²
(c) the maximum velocity of the falling tide
(a) The distance from one end of the motion to the other (3.6 m) is twice the amplitude. Therefore, A = 1.8 m.
<diagram>
Consider the point P as shown moving in a circle radius A about O with constant speed. Now consider the point Q as shown that moves up and down the vertical line between…
oscillation
any repeating motion
periodic motion
an oscillation that has the same time interval
pendulum
A device that swings back and forth due to the force of gravity
sec
units for T
they're inverses
how are frequency and period related to each other
240Hz
if f= 120Hz How many oscillations are there in 2sec
length and acceleration due to gravity
What does the period of a pendulum depend upon?
mass and spring constant
what does the period of an oscillating spring depend upon?
To increase the period...
you increase the length
natural frequency depends upon...
shape, size, and material
Resonance
a phenomenon that occurs when two objects naturally vibrate at the same frequency; the sound produced by one object causes the other object to vibrate
vibrations get larger and larger in Amplitude
what's the result of resonance, what do you achieve?
Transverse wave (ref notes)
The medium (middle) is vibrating perpendicular to the direction of the wave
longitudinal wave (ref notes)
the medium (middle) oscillates parallel to the direction of the wave
False
(T/F) Tycho Brahe was a Danish astronomer who confirmed Copernicus's theories by making many observations with his telescope.
True
Metallic bonds are formed with elements that have similar low electronegativities, true or false?
The quicker waves are produced the faster they travel.
False (if the medium is the same T is the same)
Isochronous
Having a frequency independent of amplitude.
restoring force
in a physics context, is a variable force that gives rise to an equilibrium in a physical system. If the system is perturbed away from the equilibrium, the restoring force will tend to bring…
Equilibrium Postition
The point at which there is no net force on the oscillator. For SHM this is the centre.
Elastic Strain Energy
The area under a Force vs Extension graph. SI unit: Joules
When the frequency doubles...
the maximum velocity doubles.
When the frequency halves...
the maximum acceleration quarters.
When the amplitude quadruples...
the maximum acceleration quadruples.
the time period doubles.
When the mass (on a spring) quadruples...
sound
_______ is a mechanical disturbance propagated through an elastic medium
inertia
_______ is the property of an object that leads it to resist any change in motion
elasticity
_______ is when an object returns to its initial/starting position after disturbance
vibrate
the combined properties of inertia & elasticity, causing an object to move back and forth from starting position and back again
sinusoid
_______ is the sound that results froma simple harmonic motion also known as pure tone
phase
_______ is the degree portion of a cycle at any given time
phase difference
_______ is the difference in phase angles at any given time
time
A measure of change between two events or moments.
sine wave
the waveform graph derived from simple harmonic motion is a _______
.001
1 millisecond= _______ second
pressure
_______ is the measure of force exerted on a unit area of surface
dyne
the unit of measure for pressure is the _______
watts
the unit of measure for power is _______
Power
(physics) the rate of doing work
wavelength frequency
_______ is inversely proportional to _______
.0002 2000
the dynamic range for pressure is _______ dyne/cm2 to _______ dyne/cm2
10-16 10-2
the dynamic range for power/intensity is _______ watt/cm2 to _______ watt/cm2
344
the constant for the wavelength formula is _______
amplitude time
the graph coordinates for waveform are _______ and _______
elasticity inertia
the properties of air that allow vibration are _______ and _______
brownian
Air consists of molecules in constant random motion, or _______ motion.
inertia elasticity
Air molecules have mass (_______) and spring (_______)
force/area
dyne/cm2=
energy/time/distance
watts/cm2=
reference
we use _______ because we want to compare to something that is the softest sound
0 140
the range of human hearing for dBIL is _______ to _______
loudness
amplitude is referring to the _______
pitch
frequency is referring to the _______
RMS
_______ amplitude refers to taking the square of all instantanous amplitudes, take the average of those squares, take the square root of the value of the most common measure of amplitude, particularly with the complex waveforms
344 frequency
wavelength= _______/_______
db
amplitude is measured in _______
HZ
frequency is measured in_______
degrees
Rotational Displacement can be measured in...
complex
we use RMS amplitude for _______ soun and is calculated because it is a more meaningful reference
rarefraction condensation
sound propagates through air through _______ and _______
wavlength
we can predict the frequency of sound from its distance traveled from the _______ formula
pressure intensity
the relationship between pressure and intensity is _______ squared = _______
20 to 20000HZ
the range for human hearing
Angular Velocity/Frequency
The rate of change of the angle
Time Period
The time required for one wave (T)
Angular acceleration
represented by a lower case alpha (α)
Displacement (in terms of Cosine/Sine)
x = Xcosωt (X = maximum particle displacement from mean position)
Velocity (in terms of Cosine/Sine)
v = ω(X^2 - x^2)^1/2
Examples of Oscillations
Mass on a spring (bungee jumping);
equilibrium
1)starting point of SHM, object is in _______, fnet equals zero
Displacement (for waves) (x)
The distance a particle moves in a particular direction from its mean (equilibrium) position.
Amplitude (for waves) (x₀)
Maximum displacement from the mean position
Frequency (F)
If you are given revolution per second or minute
Period (T)
the time for an object to complete one revolution
Simple Harmonic Motion (SHM)
Motion that takes place;
In phase
360 degrees or 2π off → When they overlap
Antiphase
180 degrees or π off → Frequencies cancel
out of phase
the crests of one wave overlap the troughs of another to produce regions of zero amplitude
Angular Frequency
Product of 2π and frequency;
What is harmonic motion?
When an object repeats its path at regular time intervals.
What is simple harmonic motion?
Harmonic motion that has a sinusoidal shape (sine or cosine) when the object's displacement is graphed as a function of time.
Where the object has zero net force acting on it.
What is the *equilibrium position* of an object in simple harmonic motion?
Where is the equilibrium position for a mass on a vertical spring?
Where the object would "naturally" hang because that's where the spring force and the gravitational force (weight) balance out to zero.
Where is the equilibrium position for a pendulum?
The lowest point of the swing (where the pendulum would "naturally" hang).
What is amplitude (A)?
An object's maximum displacement away from its equilibrium position.
What is the *amplitude* (A) on a graph of simple harmonic motion?
The vertical distance from the middle to the highest (peak) or the lowest (valley) point.
What is the *amplitude* for a spring?
The maximum distance the spring is stretched or compressed from its equilibrium position.
What is the *amplitude* for a pendulum?
The maximum angle the pendulum moves from its equilibrium position (the vertical).
What is period (T)?
The time it takes to complete one cycle of something (like a repeated path). In other words, the time it takes to return to the same position.
What is the *period* (T) on a graph of simple harmonic motion?
The interval between 2 similar points on the graph (for example, between 2 peaks or 2 valleys).
What is frequency (f)?
The number of cycles that occur in one second.
Hertz (Hz).
What is the unit for *frequency* (f)?
They're reciprocals.
How are *period* and *frequency* related to each other?
One back and forth oscillation.
For a spring or a pendulum, what is typically considered to be one cycle of motion?
What is the *restoring force*?
The force that acts to "restore" or return an object to its equilibrium position.
(aka, Hooke's Law)
F = restoring force
0
0
What does the negative sign in Hooke's Law (F = -kx) mean?
The negative sign reminds you that the force acts to reverse or negate the displacement.
The stiffness (of the spring).
What is the spring constant (k) sometimes called?
U = elastic potential energy
(aka, work required to stretch or compress a spring)
Is acceleration constant during simple harmonic motion?
No. The acceleration changes from a maximum at the amplitude (maximum displacement) to zero at the equilibrium position.
At maximum displacement from the equilibrium position (the amplitude).
When does a spring or a pendulum experience a maximum net force?
At the equilibrium position.
When does a spring or a pendulum experience maximum velocity?
At the amplitude (maximum displacement from the equilibrium position).
When does a spring or a pendulum have maximum potential energy?
The spring constant.
On a graph of a spring's restoring force versus its displacement, what does the *slope* of the graph represent?
The elastic potential energy. Or the work required to stretch or compress the spring.
On a graph of a spring's restoring force versus its displacement, what does the *area* under the curve represent?
T = period
period of a spring
Increase.
How will a heavier mass affect the period of a spring?
How will a heavier mass affect the period of a pendulum?
No effect. The period of a pendulum doesn't depend on mass.
Decrease.
How will a lighter mass affect the period of a spring?
How will less gravity affect the period of a spring?
No effect. The period of a spring doesn't depend on gravity.
What is damped harmonic motion?
When the amplitude decreases over time.
Displacement, x
The change of position with direction (either positive or negative) from the equilibrium
Amplitude, A
Maximum displacement
Time Period, T
The time taken for one complete cycle of an oscillation and is equal to the reciprocal of frequency (1/f). The unit is the second (s)
Frequency, f
The number of complete cycles in one second and is equal to the reciprocal of time period (1/T). The unit is the Hertz (Hz).
Angular Frequency, ω
The size of the angle turned through in one second (2π/T).
Phase Difference Equation
2 π t / T, where t is time elapsed (delta t) and T is time period
Free Oscillations
The amplitude is constant and no frictional forces are present.
Displacement-Time graph
Sinusoidal shape that cuts through the x-axis as the object pass equilibrium. The gradient indicates the velocity at that point in time.
Velocity-Time graph
Sinusoidal shape that cuts through the x-axis when the object is at maximum displacement. The gradient indicates the acceleration at that point in time.
Acceleration-Time Graph
Sinusoidal shape that cuts through the x-axis as the object pass equilibrium. Out of phase by π compare with displacement-time graph.
SHM Acceleration Equation
a = -(2 π f)² x
SHM Velocity Equation
v = ±2 π f √(A² - x²)
SHM Displacement Equation
x = A cos(2 π f t)
Spring Time Period
T = 2 π√(m/k), where m is object mass and k is spring constant
Pendulum Time Period
T = 2 π√(L/g), where L is the length of the pendulum and g is gravitational field strength
Light Damping
The time period remains constant and the amplitude gradually decreases.
Critical Damping
damping which stops the motion of an oscillating particle in minimum time.
Heavy Damping
No oscillation occurs, returns to equilibrium slowly.
Forced Oscillations
Occurs when a periodic force is applied
Periodic Force
A force that is applied at regular intervals