Some Oscillating Systems Object on a vertical spring Choose downward direction as positive Spring force on mass is -ky where y is downward displacement from unstretched position Gravity exerts force +mg Similar to

Change the variable Let y` = y-y 0 where y 0 = mg/k Then substitute y = y 0 + y`

Vertical Spring Effect of gravity is to simply shift the equilibrium position from y=0 to y`=0 ! The angular frequency is the same as for a horizontal spring ! What energy is involved? Both stretching the spring and gravitational PE

Example A 3 kg object stretches a spring by 16 cm when it hangs vertically in eqm. The spring is then stretched further from equilibrium and the object released. (a) what is the frequency of the motion? (b) what is the frequency if the 3 kg object is replaced by a 6 kg object?

Solution Ideas: f depends on force constant k and mass k can be determined from the eqm position y 0 (a) in eqm ky 0 =m 1 g substitute in

Solution(cont’d) (b) replace m 1 by m 2 =2m 1

Simple Pendulum simple pendulum : particle of mass m at the end of a massless, non-elastic string of length L what is the period T? consider the forces involved

Simple Pendulum The net force is F = -mg sin  and is tangential to the path and opposite to the displacement sin  ~  -  3 /3 + … (  in radians!) displacement along path s = L  hence for small , F ~ -mg  = -mg s/L i.e. F = - k s where k= mg/L ==> SHM for small  Recall T=2  (m/k) 1/2 for mass-spring here T=2  [m/(mg/L)] 1/2 =2  (L/g) 1/2

Measuring g We can use any pendulum to measure ‘g’ For the mass on a string T = 2  (L/g) 1/2 Plot T 2 versus L ==> T 2 = (4  2 /g)L T2T2 L slope

Natural Frequencies Any object or structure has a set of natural frequencies if we shake it at this frequency, then a large amplitude vibration occurs important factor in engineering design atoms and molecules have ‘natural’ frequencies as well

Chapter 17

Waves (107) versus Particles (105) Written on paper and ‘handed in’ -material object moves from place to place Submitted electronically by -no matter transported Same information is transported however -essentially an electromagnetic wave particle (localized in space) versus wave (extended object) neither here nor there - everywhere? How do we describe waves? Submitting an assignment

Types of Waves Mechanical waves: most familiar type -water waves, sound waves, seismic waves -all need a medium to exist Electromagnetic Waves: less familiar -visible or UV light, radio and TV waves, microwaves, x-rays, radar -can exist without a medium -speed of light in vacuum c=2.998 x 10 8 m/s Matter Waves: unfamiliar -modern technology based on these waves -electrons, protons, atoms, molecules

Waves The mathematical description is the same for all types of waves Simplest example is a wave on a stretched rope Create a pulse at one end at time t=0 The pulse travels along the rope because the rope is under tension The speed of the pulse is determined by the mass density and tension in the rope