Physics 207: Lecture 2 Notes

1
Lecture 30

• Review (the final is, to a large degree,
  cumulative)
• 50 refers to material in Ch. 1-12
• 50 refers to material in Ch. 13,14,15
• -- Chapter 13 Gravitation
• -- Chapter 14 Newtonian Fluids
• -- Chapter 15 Oscillatory Motion

• Today we will review chapters 13-15
• 1st is short mention of resonance
• Order Ch. 15 to 13

2
Final Exam Details

• Sunday, May 13th 1005am-1205pm in 125 Ag Hall
  quiet room
• Format
• Closed book
• Up to 4 8½x1 sheets, hand written only
• Approximately 50 from Chapters 13-15 and 50
  1-12
• Bring a calculator
• Special needs/ conflicts
• All requests for alternative test arrangements
  should be made by today (except for medical
  emergency)

3
Driven SHM with Resistance

• Apply a sinusoidal force, F0 cos (wt), and now
  consider what A and b do,

Not Zero!!!
4
Dramatic example of resonance

• In 1940, a steady wind set up a torsional
  vibration in the Tacoma Narrows Bridge

?
5
Dramatic example of resonance

• Eventually it collapsed

?
6
Mechanical Energy of the Spring-Mass System
x(t) A cos( ?t ? ) v(t) -?A sin(
?t ? ) a(t) -?2A cos( ?t ? ) F(t)ma(t)
Kinetic energy K ½ mv2 ½ m(?A)2
sin2(?tf) Potential energy U ½ k x2 ½
k A2 cos2(?t ?) And w2 k / m or k m
w2 K U constant
7
SHM
x(t) A cos( ?t ? ) v(t) -?A sin(
?t ? ) a(t) -?2A cos( ?t ? )

• A amplitude
• angular frequency
• 2p f 2p/T
• ? phase constant

xmax A vmax ?A amax ?2A
8
Recognizing the phase constant

• An oscillation is described by x(t) A cos(?tf).
  
• Find f for each of the following figures

Answers f 0 f p/2 x(t) A cos (p) f p
9
Common SHMs
10
SHM Friction with velocity dependent Drag force
-bv
b is the drag coefficient soln is a damped
exponential
if
11
SHM Friction with velocity dependent Drag force
-bv
If the maximum amplitude drop 50 in 10
seconds, what will the relative drop be in 30
more seconds?
12
Chapter 14 Fluids

• Density ? m/V
• Pressure P F/A P1 atm 1x105 N/m2
• Force is normal to container surface
• Pressure with Depth/Height P P0 ?gh
• Gauge vs. Absolute pressure
• Pascals Principle Same depth ? Same pressure
• Buoyancy, force, B, is always upwards
• B ?fluid Vfluid displaced g (Archimedess
  Principle)
• Flow
• Continuity Q v2A2 v1A1 (volume / time or
  m3/s)
• Bernoullis eqn P1 ½ ?v12 ?gh1 P2 ½
  ?v22 ?gh2

13
Example problem

• A piece of iron (?7.9x103 kg/m3) block weighs
  1.0 N in air.
• How much does the scale read in water?
• Solution
• In air
• T1 mg ?iromV g
• In water BT2-mg 0
• T2 mg-B
• mg ?waterVg
• mg ( ?water /?iron ) ?iron Vg
• mg (1-?water /?iron )
• 0.87mg 0.87 N

14
Another buoyancy problem

• A spherical balloon is filled with air (rair 1.2
  kg/m3). The radius of the balloon is 0.50 m and
  the wall thickness of the latex wall is 0.01 m
  (rlatex 103 kg/m3). The balloon is anchored to
  the bottom of stream which is flowing from left
  to right at 2.0 m/s. The massless string makes
  an angle of 30 from the stream bed.
• What is the magnitude of the drag force
• on the balloon?
• Key physics Equilbrium and buoyancy.
• SFx0 SFy0

15
Another buoyancy problem

• A spherical balloon is filled with air (rair 1.2
  kg/m3). The radius of the balloon is 0.50 m and
  the wall thickness of the latex wall is 1.0 cm
  (rlatex 103 kg/m3). The balloon is anchored to
  the bottom of stream which is flowing from left
  to right at 2.0 m/s. The massless string makes
  an angle of 45 from the stream bed.
• What is the magnitude of the drag force on the
  balloon?
• Key physics Equilibrium and buoyancy. SFx0
  SFy0
• SFx-T cos q D 0
• SFy-T sin q Fb - Wair 0
• Wair rair V g rair (4/3 pr3) g with r0.49 m
• Fb rwater V g rwater (4/3 pr3) g
• Variation What is the maximum wall
• thickness of a lead balloon filled with He?

Fb
D
T
Wair
16
Pascals Principle

• Is PA PB ?
• Answer No!
• Same level, same pressure, only if same fluid
  density

B
A
17
Power from a river

• Water in a river has a rectangular cross section
  which is 50 m wide and 5 m deep. The water is
  flowing at 1.5 m/s horizontally. A little bit
  downstream the water goes over a water fall 50 m
  high. How much power is potentially being
  generated in the fall?
• W Fd mgh
• Pavg W / t (m/t) gh
• Q Av and m/t rwater Q (kg/m3 m3/s)
• Pavg rwater Av gh
• 103 kg/m3 x 250 m2 x 1.5 m/s x 10 m/s2 x 50 m
• 1.8x108 kg m2/s2/s 180 MW

18
Chapter 13 Gravitation

• Universal gravitation force
• Always attractive
• Proportional to the mass ( m1m2 )
• Inversely proportional to the square of the
  distance (1/r2)
• Central force orbits conserve angular momentum
• Gravitational potential energy
• Always negative
• Proportional to the mass ( m1m2 )
• Inversely proportional to the distance (1/r)
• Circular orbits Dynamical quantities (v,E,K,U,F)
  involve radius
• K(r) - ½ U(r)
• Employ conservation of angular momentum in
  elliptical orbits
• No need to derive Keplers Laws (know the reasons
  for them)
• Energy transfer when orbit radius changes(e.g.
  escape velocity)

19
Key equations

• Newtons Universal Law of Gravity

Universal Gravitational Constant G 6.673 x
10-11 Nm2 / kg2 The force points along the line
connecting the two objects.
On Earth, near sea level, it can be shown that
gsurface 9.8 m/s2.

• Gravitational potential energy

Zero of potential energy defined to be at r
8, force ? 0

• At apogee and perigee

20
Dynamics of Circular Orbits

• For a circular orbit
• Force on m FG GMm/r2
• Orbiting speed v2 GM/r (independent of m)
• Kinetic energy K ½ mv2 ½ GMm/r
• Potential energy UG - GMm/r
• Notice UG -2 K
• Total Mech. Energy
• E KE UG - ½ GMm/r

21
Changing orbit

• A 200 kg satellite is launched into a circular
  orbit at height h 200 km above the Earths
  surface.
• What is the minimum energy required to put it
  into the orbit ? (ignore Earths spin) (ME
  5.97x1024 kg, RE 6.37x106 m, G 6.67x10-11
  Nm2/kg2)
• Solution
• Initial h0, ri RE
• Ei Ki Ui 0 (-GMEm/RE )
• -1.25x1010J
• In orbit h 200 km, rf RE 200 km
• Ef Kf Uf - ½ Uf Uf ½ Uf
• - ½ GMEm/(RE200 km)
• -6.06x109J
• DE Ef Ei 6.4x109 J

22
Escaping Earth orbit

• Exercise suppose an object of mass m is
  projected vertically upwards from the Earths
  surface at an initial speed v, how high can it go
  ? (Ignore Earths rotation)

implies infinite height
23
We hope everyone does well on Sunday

• Have a great summer!