Oscillations (from lecture 11)

periodic/oscillatory motion

motion that repeats about a point of stable equilibrium

• objs or a system of objs that undergo oscillatory motion are called oscillators

different types of position vs. time graphs for oscillating systems

amplitude

• denoted by A

• the absolute value of the max magnitude of displacement of the particle (exhibiting periodic motion) in either the positive or negative direction

• units: meters

cycle

one complete “round trip” that a particle makes when undergoing periodic motion

period

• denoted by T

• the time it takes for a particle to complete one cycle

• units: seconds

relationship between T (period) and f (frequency)

T = 1/f

simple harmonic motion

• sinusoidal oscillation

• occurs when the restoriing force is directly proportional to the displacement x from the equilibrium point → this is the case of ideal frictionless springs

the general sinusoidal equation to describe simple harmonic motion

• position function in terms of time

in what situation is the phase constant/angle of the SHM equation = 0?

if the particle is at its max position (x = amplitude) at t = 0

angular frequency for a mass-spring system

if the sinusiodal curve is shifted to the left, then the phase constant is:

positive; if shifted to the right, the phase constant is negative

how to obtain the velocity and acceleration equations from the SHM sinusoidal position function

max velocity and max acceleration of a particle in SHM

phase angle differences in the position, velocity, and acceleration functions

the reason for the phase angle is because of the differentiation of sinusoidal functions

• from trigonometry, it’s known that -sin(x) = cos(x + pi/2), and -cos(x) = cos(x + pi)

determining the phase constant from the particle’s initial speed, position, and angular freq when undergoing SHM

the last equation is the most notable

determining the amplitude from the particle’s initial speed, position, and angular freq when undergoing SHM

the last equation is the most notable

relationship between angular freq and normal freq

f = angular freq / 2*pi

formula and explanation for kinetic energy of a particle in SHM

formula and explanation for potential energy of a particle in SHM when the spring is elongated

just sub in the SHM position function for x in the elastic potential energy formula

formula and explanation for the total mechanical energy of a particle in SHM

the total mechanical energy is also equal to the maximum potential energy (since it would mean that there’s zero kinetic energy) and equal to the maximum kinetic energy (when there’s no elastic potential energy)

kinetic/potential energy graphed (vs.time and vs. displacement)

• energy is continuously being transformed between potential energy stored in the spring and the kinetic energy of a mass attached to the moving spring

formula for velocity on an arbitrary position x in SHM

the speed of a particle in SHM is at the maximum when x (displacement from equilibrium) = ____, and is zero at _______

0; the turning points +- A

damped oscillations: defn

• dissipative forces (air resistance, friction) erode the oscillations, causing them to grow smaller with time

• consequently, the mechanical energy is the system diminishes over time, and the motion is considered “damped”

retarding force that damps an the object’s periodic motion as it moves through air

position function for oscillator with some damping

angular freq of oscillation formula for an oscillator with damping

graph of displacement vs. time for a damped oscillator

this shows that the amplitude of a damped oscillator decays exponentially over time

formula for the angular freq of a damped oscillator