SIMPLE HARMONIC MOTION

STARTER

• MAKE A LIST OF OBJECTS THAT EXPERIENCE VIBRATIONS:

PERIODIC MOTION

• PERIODIC MOTION: Any motion that repeats itself precisely over equal periods of time.

• If that periodic motion is generated by a linear restoring force it is SIMPLE HARMONIC MOTION.

HOOKE’s LAW• Recall Hooke’s Law:

F= -kx• x is the amount of extension or compression of

a spring• Spring constant, k , describes the force

necessary to stretch or compress an individual spring. Each spring is different.

• the negative sign indicates that the force always opposes the direction of the stretch, “restoring” the object to its rest or equilibrium position

Quantities that Describe Periodic Motion

• Period: the length of time required to complete one full cycle of motion

• Frequency: the number of cycles in a given time

• Period and frequency are inversely related

ENERGYET= ½ mv2 + ½ kx2

• x is the position• At the ends of the

oscillation the mass stops and changes direction, so all of the energy is elastic potential energy

• At equilibrium (x=0) the string is not stretched so all energy is kinetic.

VELOCITYv = vmax √ 1- x2/A2

• v is the speed of the mass at any position of its motion

• x is any position• A is maximum amplitude• vmax is the maximum

speed; occurs at equilibrium position

UNIFORM CIRCULAR MOTION

• The amplitude of SHM is equal to the radius of Uniform Circular Motion

• Use the UCM equations to describe SHM:a = 4π2A/T2

vmax = 2πA/ T Or T = 2πA/ vmax

PERIOD and FREQUENCY of mass on spring

• Another arrangement for vmax

vmax2 = k/m A2

• Substituting into UCM equationT = 2π√m/k

• Equation for frequencyf = 1/2π √k/m

PENDULUM

T =2π √ l /g• l is the length of the pendulum string• g is acceleration due to gravity• This equation is valid for the period of a

pendulum for small angles