Some Oscillating SystemsObject on a vertical spring

Choose downward direction as positive

Spring force on mass is -kywhere y is downward displacementfrom unstretched position

Gravity exerts force +mg2

2

d ym ky mgdt

2

2

d xm kxdt

Similar to

Change the variable

• Let y` = y-y0 where y0 = mg/k

• Then substitute y = y0 + y`

20

02

( ` )( ` )

d y ym k y y mg

dt

2

2

d ym ky mgdt

2

2

``

d ym kydt

` cos( )y A t

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

/k m

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 y0

• (a)

• in eqm ky0=m1g

• substitute in

1

1/

2 2f k m

1

0

(3 )(9.81 / )184 /

.16

m g kg N kgk N m

y m

21 0

11

/1 1 1 9.81 / )/ 1.25

2 2 2 .16

m g y m sf k m Hz

m m

Solution(cont’d)

• (b) replace m1 by m2=2m1

2 2 1

1 1 1.25/ .884

2 2 2 2

Hzf k m f Hz

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 T2 versus L ==> T2 = (4 2/g)L

T2

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

17 1, 3, 6, 9 1, 2, 7, 10, 14, 18, 21, 26, 40, 42 3, 6, 7

Waves (107) versus Particles (105)

• Written on paper and ‘handed in’ -material object moves from place to place

• Submitted electronically by email -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 108 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