Simple Harmonic Motion Examples of Periodic Motion Swinging on a playground Guitar string Metal block bobbing up and down Pendulum

Periodic Motion One position where Fnet = 0 = equilibrium Whenever the object is out of equilibrium, the Fnet pulls back If F that restores the equilibrium is directly proportional to displacement of object SIMPLE HARMONIC MOTION

At equilibrium, speed is MAX See next Figure, A) spring is stretched away from its unstretched (equilibrium) position (x=0) Spring exerts a force on the mass towards the equilibrium position Spring force and acceleration decreases as the mass moves towards eq. position B) Zero force at eq.; mass acceleration is zero Yet speed is increasin towrds eq. andreaches MAXIMUM at eq. Fsp=0; mass inertia causes it to overshoot eq. and compress spring

At max displacement, Fsp and a reach a MAX C) Mass moves past eq, Fsp and s increase Direction of Fsp and a (towards eq.) is opposite the mass’s dorection of motion Mass slows down When spring compression equals the distance of original stretch (displacement) Mass is at max displacement Fsp and a are at a max Mass speed = o , force causes mass to change direction Repeat oscillation (ignore resistance forces) Damping: real world oscillation (resistance forces)

Hooke’s Law In SHM, restoring force is proportional to displacement. Fsp is called restoring force since it is always returning the system to equilibrium (Fnet =0) F elastic = -kx Spring force = - (spring constant x displacement) (-) sign signifies that the direction of the spring force is always opposite the displacement Higher k means a stiffer spring K has units N/m At Eq, Fnet = 0 = F elastic + Fg

Sample Problem If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant? Given: m=0.55 kg x=-2.0cm= -0.020m G = -9.8m/s/s Unknown is k=? Felastic Fg

Simple Harmonic Motion is described by 2 traits Period (T): the time needed to repeat one complete cycle of motion Amplitude: the maximum distance moved from equilibrium

Mass in a Spring Period of oscillation depends on the mass of the block and strength of the spring Fsp = F g in equilibrium

Simple Pendulum: bob (mass) attached to a string A pendulum moves with SHM if the angle is less than 15 degrees. Fnet is opposite and linear to displacement Period of a pendulum does not depend on mass or amplitude http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c1 FT Fnet Fg

Restoring Force of a Pendulum is a component of the bob’s weight Force of string along the y axis Force of gravity can be resolved into comonents Net force if the x component of Fg Since Fgx is towards the equilibrium position it is the restoring force Frestoring Fg,x = Fgsin theta Restoring force is zweo at eq. Small anfles this acts in simple harmonic motion FT F gx Fgy

See SMH Figure

Measuring SHM Term Example Definition SI unit Amplitude Maximum displacement from equilibrium Radian, rad Meter, m Period , T Time that it takes to complete a full cycle Secons, s Frequency, f Number of cycles or vibrations per unit of time Hertz, Hz (Hz=s-1)

Formulas for SHM f = 1/T or T = 1/ f Ex. If one complete cycle takes 20 s T = 20 s frequency: 1/20 cycles /s or 0.05 cycles /s 0.05 Hz

Formulas and sample problem Period of a simple Pendulum in SHM You need to know the height of a tower, but darkness obscures the ceiling. You note that a pendulum extedning from the ceiling almost touches the floor and that its period is 12 s. How tall is the tower? (answer L=36 m)

Mechanical Resonance Small forces are applied at regular intervals to a vibrating or oscillating object to increase the amplitude Examples Jumping on a diving board Jumping on a trampoline

SHM: CFU What is the length of a pendulum with a period of 1.00 s? (0.248 m) 2. Would it be practical to make a pendulum with a period of 10.0 s? Calculate the length and explain.(24.8 m; No, this is over 75 ft long!) 3. On a planet with an unknown value of g, the period of a 0.65 m long pendulum is 2.8 s. What is the g for this planet? (3.3 m/s2)