Simple Harmonic Motion Syll. State. 4.1.1-4.2.3 SS/Note template due next Monday (get note template...
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Transcript of Simple Harmonic Motion Syll. State. 4.1.1-4.2.3 SS/Note template due next Monday (get note template...
Simple Harmonic MotionSyll. State. 4.1.1-4.2.3
SS/Note template due next Monday (get note template from the website)
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…
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
Magnitude of Force…
Depends on the displacement from equilibrium
Always (ALWAYS) is in the direction pointing toward the equilibrium point
Hooke’s Law:
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)
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
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
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
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
Frequency vs. Period
Frequency and period are inverses of each other.
Period is the time needed per cycle (or oscillation)—measured in sec.
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
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: