Simple Harmonic MotionReminders:

Syll. State. 4.1.1-4.1.8 Due Tuesday

WA due Tuesday

Quiz Tuesday

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

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:

Wait…where’d that equation come from?

Newton’s 2nd law: Hooke’s Law: Angular frequency defined: Put it together:

Oscillations—position vs timeWhat pattern do we see when we plot

a graph of position vs. time for an oscillating mass?

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