Review for the third test

1
Review for the third test

• Goals

• Review Chapters 14-17

• Third test on Thursday at 715 pm.

2
Chapter 14, SHM
k
T2? (m/k)½ T2? (L/g)½ for pendulum
m
-A
A
0(Xeq)
The general solution is x(t) A cos ( wt
f)
with w (k/m)½ and w 2p f 2p /T
T 2?/?
A
-A
3
Energy of the Spring-Mass System
x(t) A cos( ?t ? ) v(t) -?A sin( ?t ?
) a(t) -?2A cos( ?t ? )
Potential energy of the spring is U ½ k x2
½ k A2 cos2(?t ?) The Kinetic energy is
K ½ mv2 ½ k A2 sin2(?tf)
4
Damped oscillations

• For small drag (under-damped) one gets

x(t) A exp(-bt/2m) cos (wt f)
x(t)
t
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Exercise
x (m)
0.2
t (s)
4
2
3
1
-0.2

• What are the amplitude, frequency, and phase of
  the oscillation?

x(t) A cos ( wt f)
w 2p f p rad/s
fp rad
6
Chapter 15, fluids
PressureP0
FP0AMg
P0A
PP0?gy
y
AreaA
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Pressure vs. Depth

• In a connected liquid, the pressure is the same
  at all points through a horizontal line.

p
8
Buoyancy
F2P2 Area F1P1 Area F2-F1(P2-P1)
Area ?g(y2-y1) Area ? g Vobject weight
of the fluid displaced by the object
y1
F1
y2
F2
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Pascals Principle
Any change in the pressure applied to an enclosed
fluid is transmitted to every portion of the
fluid and to the walls of the containing vessel.
Hydraulics, a force amplifier
P1 P2 F1 / A1 F2 / A2 A2 / A1
F2 / F1
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Exercise
m2? kg r21 m
m110 kg r10.1 m
P1 P2 m1g / A1 m2g / A2 m2m1
A2/A11000 kg
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m2? kg r21 m
m110 kg r10.1 m
h
A

• What is the pressure at the bottom of the
  container?

PA1 atm m1g/ A1?gh
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Continuity equation
A1v1 units of m2 m/s volume/s A2v2
units of m2 m/s volume/s
A1v1A2v2
13
Energy conservation Bernoullis eqn
P11/2 ? v12?gy1P21/2 ? v22?gy2
P1/2 ? v2?gy constant
14
Chapter 16, Macroscopic description

• Ideal gas law

P V n R T P V N kB T
n of moles N of particles
15
PV diagrams Important processes

• Isochoric process V const (aka
  isovolumetric)
• Isobaric process p const
• Isothermal process T const

16
Work done on a gas
W - the area under the P-V curve
Pi
Pressure
Pf
Vi
Vf
Volume
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Chapter 17, first law of Thermodynamics
?U?K?Ethermal?EsystemWexternalQ

• For systems where there is no change in
  mechanical energy

?Ethermal WexternalQ
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Exercise
Pressure
i
f
Pi
Volume
Vi
3Vi

• What is the final temperature of the gas?
• How much work is done on the gas?

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Thermal Properties of Matter
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Heat of transformation, specific heat

• Latent heat of transformation L is the energy
  required for 1 kg of substance to undergo a phase
  change. (J / kg)
• Q ML

• Specific heat c of a substance is the energy
  required to raise the temperature of 1 kg by 1 K.
  (Units J / K kg )
• Q M c ?T

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Specific heat for gases

• For gases we typically use molar specific heat
  (Units J / K mol )
• Q n C ?T

• For gases there is an additional complication.
  Since we can also
• change the temperature by doing work, the
  specific heat depends
• on the path.
• Q n CV ?T (temperature change at constant V)
• Q n CP ?T (temperature change at constant P)

CPCVR
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Exercise
Pressure
i
f
Pi
Volume
Vi
3Vi

• How much heat energy is transferred to or from
  the gas?