Week 5 Free and Forced Vibrations

oscillatory motion

the repetitive back-and-forth movement of an object around a central equilibrium point (ex. pendulum); are all perodic motion

periodic motion

any movement that repeats itself over equal time intervals (ex. clocks); are not all oscillatory motion

finding the period of the pendulum

T=2pi x sqrt(L/g)

T=period, L=length of string,g=gravitational acceration (9.81 m/s)

does mass of the pendulum affect the period?

no

if you displace the pendulum one small and one larger, which would oscillate longer?

the displacement doesn’t affect the period (displacement isn’t a variable to find the period)

angular frequency

the angular displacement (change in angle) of any element of the wave per unit time (rad/period)f

equation to find phase difference in radians

2pi(change in t)/T

free oscilation

no restrictions, constant amplitude

damped oscillations

any oscillating system where friction or air resistance is present, the amplitude decreases

types of damped oscillations

critical damping, light damping, and heavy damping

forced vibrations

aka forced oscillations, when a periodic force is applied to it (ex. pushing someone on a swing)

natural frequency

frequency (rate the system vibrates) when disturbed from resting position, w/o external driving or damping forces (instead an intial input of energy, like a flick)

critical damping

optimal level of damping that allows it to return to equilibrium in the shortest time without oscilating (ex.bouncing) at all

light damping

where amplitude of oscilations decreases gradually over time, allowing it to oscillate for a longer duration

heavy damping

system returns to its equilibrium position very slowly without oscilating, typically due to strong resistive forces (like those cabinets that doesn’t close loudly)

simple harmonic motion

oscillatory motion described by just one sine or cosine function; where restoring force is directly proportional to the displacement, related to Hooke’s law

hookes law

states that force required to extend/compress a spring by a certain distance is directly propertional to that distance, as long as the elastic limit is not exceeded F=kx

elastic potential energy

potential energy in an elastic object when strethced or compressed

elastic potential energy is proportional/not proportional to the dispacement

proportional, as Ep=1/2kx²

the elastic potential energy in simple harmonic motion is at its minimum

when the spring is in its equilibrium position

restoring force

the force that acts to bring a system back to its equilibrium position when displaced; proportional to displacement, described in Hooke’s law

harmonic oscillator

physical system that undergoes simple harmonic motion

harmonic motion

oscillatory motion described as a combination of sine and/or cosine functions; where restoring force is approximately proportional to the displacement (simple harmonic motion is a type of harmonic motion)

why is hooke’s law for restoring force F=-kx?

the force of the springs always opposite of the direction of movement

acceleration

velocity/time

mechanical energy at different points of the harmonic motion

@ max displacement = kinetic energy, 2 equilibrium = potential energy, anywhere in between = kinetic + potential energy