Some Oscillating Systems

1
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
2
Change the variable

• Let y y-y0 where y0 mg/k
• Then substitute y y0 y

3
Vertical Spring

• Effect of gravity is to simply shift the
  equilibrium position from y0 to y0 !
• The angular frequency is
• the same as for a horizontal spring !
• What energy is involved? Both stretching the
  spring and gravitational PE

4
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?

5
Solution

• Ideas
  f depends on force
  constant k and mass
• k can be determined from the eqm position y0
• (a)
• in eqm ky0m1g
• substitute in

6
Solution(contd)

• (b) replace m1 by m22m1

7
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

8
(No Transcript)
9
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
• gt SHM for small ?
• Recall T2 ?(m/k)1/2 for mass-spring
• here T2 ?m/(mg/L)1/2 2 ?(L/g)1/2

10
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 gt T2 (4 ?2/g)L

T2
slope
L
11
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

12
(No Transcript)
13
Chapter 17
14
Waves (107) versus Particles (105)
Submitting an assignment

• 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?

15
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 c2.998 x 108 m/s
• Matter Waves unfamiliar -modern technology
  based on these waves -electrons, protons,
  atoms, molecules

16
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 t0
• 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