REVIEW & SUMMARY OSCILLATIONS
Frequency The frequenry f of periodic or oscillatory motion is the number of oscillations per second. In the SI system, it is measured in hertz I hertz = I Hz = I oscillation per second = I s -I Period The period T is the time required for one complete oscillation or cycle It is related to the frequency by
Simple Harmonic Motion In simple harmonic motion (SHM) the displacement x(t) of a particle from its equilibrium position is described by the equation displacement in which I’m is the amplitude of the displacement the quantity is the phase of the motion and d is the phase constant The angular frequency W is related to the period and frequency of the motion b
Differentiating leads to equations for the panicle’s velocity and acceleration during SHM as functions of time and l’ = -WX., sin(wl + rjJ)
a = -w2x., cos(W! + rjJ) velocity acceleration In the positive quantity is the velocity amplitude of the motion. In the positive quantity is the acceleration amplitude am of the motion In the positive quantity is the velocity amplitude l’m of the motion In the positive quantity is the acceleration amplitude am of the motion.
The Linear Oscillator A panic with mass that moves under the influence of a Hooke’s law restoring force given by exhibits simple harmonic motion with
Such a system is called a linear simple harmonic oscillator Energy A particle in simple harmonic motion has at any time kinetic energy and potential energy If no friction is present, the mechanical energy E = K + U remains constant even though K and U change Pendulums Examples of devices that undergo simple harmonic motion are the torsion pendulum of the simple pendulum of and the physical pendulum of Their periods of oscillation fer small oscillations are respectively.
Simple Harmonic Motion and Uniform Circular Motion Simple harmonic motion is the projection of uniform circular onto the diameter of the circle in which the latter motion occurs that all parameters of circular motion position velocity and acceleration project to the corresponding values for simple harmonic motion Damped Harmonic Motion The mechanical energy E in a real oscillating system decreases during the oscillations because external forces such as a drag force inhibit the oscillations and transfer mechanical energy to thermal energy. The real oscillator and its motion are then said to be damped. If the damping force is given where v is the velocity of the oscillator and b is a damping constant, then the displacement of the oscillator is given by
where we the angular frequency of the damped oscillato is given by
If the damping constant is small is the angular frequency of the un damped oscillator. For b the mechanical energy E of the oscillator given by Forced Oscillations and Resonance If an external force with angular frequency acts on an oscillating system natural angular frequency the system oscillates with frequency The velocity amplitude of the system is when a condition called resonance. The amplitude Xm of the system(approximately) greatest under the same condition.