.MCAD 304020000 1 74 150 0
.CMD PLOTFORMAT
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 0 0 3 
.CMD FORMAT  rd=d ct=10 im=i et=3 zt=15 pr=3 mass length time charge temperature tr=0 vm=0
.CMD SET ORIGIN 0
.CMD SET TOL 0.001000000000000
.CMD SET PRNCOLWIDTH 8
.CMD SET PRNPRECISION 4
.CMD PRINT_SETUP 1.200000 1.218750 1.200000 1.200000 0
.CMD HEADER_FOOTER 1 1 *empty* *empty* *empty* 0 1 *empty* *empty* *empty*
.CMD HEADER_FOOTER_FONT fontID=14 family=Arial points=10 bold=0 italic=0 underline=0 colrid=-1
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.CMD DEFAULT_TEXT_PARPROPS 0 0 0
.CMD DEFINE_FONTSTYLE_NAME fontID=0 name=Variables
.CMD DEFINE_FONTSTYLE_NAME fontID=1 name=Constants
.CMD DEFINE_FONTSTYLE_NAME fontID=2 name=Text
.CMD DEFINE_FONTSTYLE_NAME fontID=4 name=User^1
.CMD DEFINE_FONTSTYLE_NAME fontID=5 name=User^2
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.CMD DEFINE_FONTSTYLE_NAME fontID=8 name=User^5
.CMD DEFINE_FONTSTYLE_NAME fontID=9 name=User^6
.CMD DEFINE_FONTSTYLE_NAME fontID=10 name=User^7
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.CMD UNITS U=1
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.TXT 1 2 146 0 0
Cg a72.000000,72.000000,28
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24\b \pard {\fs20 
COPYRIGHT MIRON KAUFMAN 1997}}
.TXT 4 18 77 0 0
Cg a54.000000,54.000000,45
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24\b \pard {\fs32 
Thermal Physics PHY474\par        Computer Lab #3}}
.TXT 7 -19 86 0 0
Cg a72.250000,72.250000,1361
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}{\f1\fcharset2\fnil Symbol;}}\plain
\cf1\fs24\b \pard {\b0 \tab We simulate a mechanical experiment used 
to measure the constant}{\f1\b0 g}{\b0  = C}{\b0\dn p}{\b0 /C}{\b0\dn V}{
\b0 . This method was developed by Ruchardt in 1929.  The gas is in a 
container of volume V}{\b0\dn 0}{\b0 .  A tube of cross-sectional area 
A in which a ball of mass m fitts perfectly is part of the container 
as shown in the Figure. The ball is in equilibrium when: }{\b0 \par }{
\b0 \tab mg + A}{\b0 p}{\b0\dn atm}{\b0  = Ap}{\b0\dn 0}{\b0 .\par The  
falling ball compresses the gas which then expands and pushe}{\b0 s 
the ball up.  The oscillations are fast enough for heat not to be 
exchanged, i.e.  adiabatic process.   We start by writting Newton's 
second law for the ball: \par \tab md}{\b0\up 2}{\b0 z/dt}{\b0\up 2 }{
\b0 = -mg - Ap}{\b0\dn atm}{\b0  + Ap.\par Assuming the gas to be 
ideal and the process adiabatic, we can use the Poisson equation:   pV}{
\f1\b0\up g}{\b0  = p}{\b0\dn 0}{\b0 V}{\b0\dn 0}{\f1\b0\up g}{\f1\b0 
.  T}{\b0 aking z = 0 when the ball is in equilibrium,  if the ball is 
above the equilibrium position then the gas volume is V = V}{\b0\dn 0}{
\b0  + Az.  }{\f1\b0 T}{\b0 he differential equation for z(t) becomes:  }{
\b0 \tab \tab d}{\b0\up 2}{\b0 z/dt}{\b0\up 2 }{\b0 = -g  + (A/m)[p}{
\b0\dn 0}{\b0 (1 + Az/V}{\b0\dn 0}{\f1\b0 )}{\b0\up -}{\f1\b0\up g}{\b0  
- p}{\b0\dn atm}{\b0 ],  where }{\b0 p}{\b0\dn 0 }{\b0 = }{\b0 p}{\b0
\dn atm }{\b0 + }{\b0 mg/A.\par If the oscillations are small we can 
expand the right hand side of the differential equation in powers of 
z:\par \tab }{\b0 d}{\b0\up 2}{\b0 z/dt}{\b0\up 2}{\b0  + }{\f1\b0 w}{
\b0\up 2}{\b0 z = 0 where the angular velocity is:  }{\f1\b0 w}{\b0  = [(A}{
\b0\up 2}{\f1\b0 g}{\b0 p}{\b0\dn 0}{\b0 )/(}{\b0 V}{\b0\dn 0}{\b0 m)]}{
\b0\up 1/2}{\b0  .\par So by measuring the period of small 
oscillations and by using T = 2}{\f1\b0 p}{\b0 /}{\f1\b0 w}{\b0  one 
can determine the constant }{\f1\b0 g:\par \tab }{\f1\b0 g}{\b0  = 4}{
\f1\b0 p}{\b0\up 2}{\b0 mV}{\b0\dn 0}{\b0 /[T}{\b0\up 2}{\b0 A}{\b0\up 2}{
\b0 (p}{\b0\dn atm}{\b0 +mg/A)].\par We start by inputing the 
information about the ball (m, A) and the gas (V}{\b0\dn 0 }{\b0 ,p}{
\b0\dn atm }{\b0 ,}{\f1\b0 g}{\b0 )}{\b0 .  }}
.EQN 56 5 64 0 0
{0:m}NAME:(10)^(-2)
.EQN 0 10 65 0 0
{0:A}NAME:(10)^(-4)
.EQN 0 13 66 0 0
{0:\g}NAME:(5)/(3)
.EQN 3 -23 67 0 0
{0:g}NAME:9.8
.EQN 0 10 68 0 0
{0:V.0}NAME:(10)^(-2)
.EQN 0 11 69 0 0
{0:p.atm}NAME:(10)^(5)
.EQN 4 -21 95 0 0
{0:p.0}NAME:{0:p.atm}NAME+{0:m}NAME*({0:g}NAME)/({0:A}NAME)
.TXT 4 -4 99 0 0
Cg a72.000000,72.000000,65
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24\b \pard The period 
computed from the small oscillations approximation is:}
.EQN 5 4 96 0 0
{0:T}NAME:2*{0:\p}NAME*\(({0:V.0}NAME*{0:m}NAME)/(({0:A}NAME)^(2)*{0:p.0}NAME*{0:\g}NAME))
.EQN 0 22 97 0 0
{0:T}NAME={0}?_n_u_l_l_
.TXT 6 -25 100 0 0
Cg a71.000000,71.000000,111
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24\b \pard We solve the 
differential equation by using the 4'th order Runge-Kutta method.  The 
time interval is 0 to ttot.}
.EQN 5 2 101 0 0
{0:N}NAME:1000
.EQN 2 12 106 0 0
{0:ttot}NAME:{0:N}NAME*({0:T}NAME)/(200)
.EQN 2 -12 107 0 0
{0:n}NAME:0;{0:N}NAME
.TXT 8 -1 108 0 0
Cg a70.000000,70.000000,49
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24\b \pard The initial 
values of position and velocity are:\par }
.EQN 0 36 131 0 0
{0:\z}NAME:({2,1}ö{0}0ö100)
.TXT 0 10 143 0 0
Cg a24.000000,24.000000,114
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24\b \pard Note: to 
study the effect of the nonlinearity in force it helps to input a very 
large (and unrealistic) initial z.}
.TXT 5 -46 129 0 0
Cg a70.000000,70.000000,109
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}{\f1\fcharset2\fnil Symbol;}}\plain
\cf1\fs24\b \pard The first derivative of z is velocity:{\f1   }d{\f1 z}{
\f1\b0\dn 0}/dt =  {\f1 z}{\f1\b0\dn 1}; \par The second derivative is 
acceleration:{\f1  }d{\f1\b0\up 2}{\f1 z}{\f1\b0\dn 0}{\f1 /}dt{\f1\b0
\up 2}{\f1  }= F/m.}
.EQN 11 34 133 0 0
{0:\Z}NAME({0:t}NAME,{0:\z}NAME):({2,1}ö-{0:g}NAME+({0:A}NAME)/({0:m}NAME)*({0:p.0}NAME*((1+{0:A}NAME*(({0:\z}NAME)[(0))/({0:V.0}NAME)))^(-{0:\g}NAME)-{0:p.atm}NAME)ö({0:\z}NAME)[(1))
.TXT 7 -36 134 0 0
Cg a72.000000,72.000000,183
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24\b \pard We call now 
the rkfixed function provided by Mathcad.  You should get more 
information on this function by using: Help?, Functions, Built-in 
functions, Differential equations solvers.}
.EQN 7 1 135 0 0
{0:z}NAME:{0:rkfixed}NAME({0:\z}NAME,0,{0:ttot}NAME,{0:N}NAME,{0:\Z}NAME)
.TXT 4 0 137 0 0
Cg a71.000000,71.000000,436
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24\b \pard z is a 
matrix: N+1 x 3. The first column contains the time values; the second 
column contains the position z values; the third column contains the 
velocity values.\par We graph first: position as function of time.  We 
can estimate the period of the oscillations. Note it is longer than T, 
the period for small oscillations.  Also note that the average value 
of z is positive, the ball spends most of the time above the 
equilibrium position.}
.EQN 11 -2 81 0 0
&&(_n_u_l_l_&_n_u_l_l_)&({0:z}NAME)[({0:n}NAME,1)@&&({0:T}NAME&2*{0:T}NAME)&({0:z}NAME)[({0:n}NAME,0)
0 1 1 1 1 0 1 1 1 
0 1 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 60 41 10 0 3 
.TXT 53 0 148 0 0
Cg a72.000000,72.000000,28
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24\b \pard {\fs20 
COPYRIGHT MIRON KAUFMAN 1997}}
.TXT 4 0 145 0 0
Cg a70.000000,70.000000,35
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24\b \pard Next we 
graph velocity versus time.}
.EQN 1 0 144 0 0
&&(0&_n_u_l_l_)&({0:z}NAME)[({0:n}NAME,2)@&&({0:T}NAME&2*{0:T}NAME)&({0:z}NAME)[({0:n}NAME,0)
0 1 1 1 1 0 1 1 1 
0 1 1 1 1 0 1 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 60 31 10 0 3 
.TXT 47 3 140 0 0
Cg a70.000000,70.000000,53
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24\b \pard Finally we 
get the trajectory in phase space (z,v).  }
.EQN 4 -3 149 0 0
&&(_n_u_l_l_&_n_u_l_l_)&({0:z}NAME)[({0:n}NAME,2)@&&(_n_u_l_l_&_n_u_l_l_)&({0:z}NAME)[({0:n}NAME,1)
0 1 1 1 1 0 0 1 1 
0 1 1 1 1 0 0 1 1 
0 0 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 61 37 10 0 3 
.TXT 48 1 150 0 0
Cg a72.000000,72.000000,28
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24\b \pard {\fs20 
COPYRIGHT MIRON KAUFMAN 1997}}
.EQN 3 4 57 0 0
{0:x}NAME:-0.999,-.998;2
.EQN 2 6 56 0 0
10&&(_n_u_l_l_&_n_u_l_l_)&{0:x}NAME+(((1+{0:x}NAME))^(1-{0:\g}NAME))/({0:\g}NAME-1)@&&(_n_u_l_l_&_n_u_l_l_)&{0:x}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3