Simple Harmonic Motion Reminders: Syll. State. 4.1.1-4.1.8 Due Tuesday WA due Tuesday Quiz Tuesday.
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Transcript of Simple Harmonic Motion Reminders: Syll. State. 4.1.1-4.1.8 Due Tuesday WA due Tuesday Quiz Tuesday.
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Simple Harmonic MotionReminders:
Syll. State. 4.1.1-4.1.8 Due Tuesday
WA due Tuesday
Quiz Tuesday
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Oscillations—what are they?Repetitive, cyclical motion in which a mass
(particle) moves back and forth around a single fixed point with a regular frequency
A.k.a Harmonic motion, or periodic motion
Examples:The “sting” of a ball hitting a bat
Strings on a violin that is being bowed
The swaying of buildings in wind or in earthquakes
And many, many more…
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So…what causes oscillations?
Restoring Force: When a particle is displaced from its equilibrium
position, it wants to return to that point
The force applied to a particle in order to bring it back to its equilibrium is called the restoring force
When the restoring force varies at a regular rate from + Fmax to – Fmax and back again, the object is oscillating due to this restoring force
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Magnitude of Force…
Depends on the displacement from equilibrium
Always (ALWAYS) is in the direction pointing toward the equilibrium point
Hooke’s Law:
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Simple Harmonic Motion
Defined by the way a mass oscillates around a fixed point
The restoring force acting on the mass is non-constant
Force acting on, and therefore, acceleration of, the mass are proportional to the displacement of the mass (Hooke’s law)
Defining equation:
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Wait…where’d that equation come from?
Newton’s 2nd law: Hooke’s Law: Angular frequency defined: Put it together:
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Oscillations—position vs timeWhat pattern do we see when we plot
a graph of position vs. time for an oscillating mass?
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Simple Harmonic Motion (SHM)Specific oscillatory behavior in which the
object oscillating follows a pattern that is a sinusoidal function of time:
Variables:
X(t) = position at time t
Xm = amplitude (maximum displacement)
w = angular frequency (rad∙s-1)
f = phase constant (rad)
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Let’s define those variables a bit more:
Displacement: The position, measured from the equilibrium point, of the particle at any time t in its oscillation
Amplitude: the maximum displacement of a particle from its equilibrium position
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Angular Frequency vs. Frequency
Frequency: the rate at which oscillations occur. Measured by counting the number of times an oscillating particle passes by a fixed point each second. units = s-1 (or, cycles per second)
Angular Frequency: the rate at which oscillations pass through the radian measure of an oscillation.Typically—units are in radians per second
(rad∙s-1)
1 oscillation = 2p radians
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Frequency and angular frequency… quantified
Frequency (f), measured in Hertz (Hz) or sec-1 Angular frequency (w), measured in rad∙s-1
𝜔=2𝜋 𝑓
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What will cause the frequency to change?
Frequency of an oscillating mass…Does NOT depend on the amplitude
DOES depend on the spring constant
DOES depend on the mass
𝜔=√ 𝑘𝑚
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Frequency vs. Period
Frequency and period are inverses of each other.
Period is the time needed per cycle (or oscillation)—measured in sec.
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Phase Constant, f
The phase constant is a value given to show at what point in the oscillation the timer had begun.
In other words, at what radian position was the oscillating mass at time t = 0 sec.?
Units = radians
Similarly, Phase difference is the difference in radian position at time t=0 for 2 waves or oscillating masses