Simple Harmonic Motion

Simple Harmonic Motion Syll. 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

What will cause the frequency to change? • Frequency of an oscillating mass… • Does NOT depend on the amplitude • DOES depend on the springconstant • DOES depend on the mass

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:

Wait…where’d that equation come from? • Newton’s 2nd law: • Hooke’s Law: • Angular frequency defined: • Put it together:

Simple Harmonic Motion