class14 - PHYS 5900 Class 14 (9/25/2008Fri) Zi-Wei Lin...

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Unformatted text preview: PHYS 5900 Class 14 (9/25/2008Fri) Zi-Wei Lin Example 2.4.13 Plot the x-y trajectory of a plane pendulum with a radius r by choosing a proper unit for time; the pendulum starts from the angle of 12 degrees at rest at time 0. Θ r Force Looking at the force and its direction at angle theta , the equation of motion is m r d 2 theta d tt 2 = - m * g * sin H theta L i.e. d 2 theta d tt 2 = - g r * sin H theta L where tt represents time, g is the gravitational acceleration of the Earth. Mathematica cannot solve it analytically : In[1]:= DSolve @8 theta '' @ tt D- g r * Sin @ theta @ tt DD , theta @ D 12 * Degree, theta ' @ D == < , theta @ tt D , tt D Out[1]= 8< Now we solve it numerically. In the above equations, I tt 2 * g r M is dimensionless and can be defined as a new variable t 2 , i.e., we choose the unit of tt as r g and then get d 2 theta dt 2 = - sin H theta L In[2]:= DSolve @8 theta '' @ t D- Sin @ theta @ t DD , theta @ D 12 * Degree, theta ' @ D == < , theta, t D Out[2]= 8< In[3]:= NDSolve @8 theta '' @ t D- Sin @ theta @ t DD , theta @ D 12 * Degree, theta ' @ D == < , theta, 8 t, 0, Pi <D Out[3]= 88 theta fi InterpolatingFunction @88 0., 3.14159 << , <> D<< In[4]:= sol = theta . First @ % D Out[4]= InterpolatingFunction @88 0., 3.14159 << , <> D In[5]:= xyList = 8 Sin @ sol @ t DD ,- Cos @ sol @ t DD< ; In[6]:= Plot @ xyList, 8 t, 0, Pi <D Out[6]= 0.5 1.0 1.5 2.0 2.5 3.0- 1.0- 0.8- 0.6- 0.4- 0.2 0.2 In[7]:= traj = Table @ xyList, 8 t, 0, Pi, Pi 8 <D Out[7]= 88 0.207912,- 0.978148 < , 8 0.192402,- 0.981316 < , 8 0.147937,- 0.988997 < , 8 0.0807484,- 0.996735 < , 8 0.000902549 ,- 1. < , 8- 0.079086,- 0.996868 < , 8- 0.146674,- 0.989185 < , 8- 0.19172,- 0.98145 < , 8- 0.207904,- 0.978149 << 2 class14.nb In[8]:= ListPlot @ traj, AxesLabel-> 8 " x H r L ", "y H r L " <D Out[8]=- 0.2- 0.1 0.1 0.2 x H r L- 0.995- 0.990- 0.985- 0.980 y H r L In[9]:= ListPlot @ traj, AxesLabel-> 8 " x H r L ", " y H r L " < , AspectRatio-> Automatic, PlotRange-> 8 0,- 1.1 <D Out[9]=- 0.2- 0.1 0.1 0.2 x H r L- 1.0- 0.8- 0.6- 0.4- 0.2 y H r L class14.nb 3 We can also use ParametricPlot : In[10]:= ParametricPlot @ xyList, 8 t, 0, Pi < , PlotRange fi 8 Automatic, 8 0,- 1.1 <<D 4 class14.nb Out[10]=- 0.2- 0.1 0.1 0.2- 1.0- 0.8- 0.6- 0.4- 0.2 In[11]:= Clear @ sol, traj D class14.nb 5 Example 2.4.14 1) Prove the following Jacobi identity: [A×(B×C)]+[B×(C×A)]+[C×(A×B)]=0. Vector analysis: the package is "VectorAnalysis`". Functions include: CoordinateSystem, Coordinates, SetCoordinates, DotProduct, CrossProduct, Curl, Div, Grad,Laplacian, . (for DotProduct) In[12]:= Needs @ "VectorAnalysis`" D In[13]:= CoordinateSystem Out[13]= Cartesian In[14]:= Coordinates @D Out[14]= 8 Xx, Yy, Zz < In[15]:= Div @8 Xx^2, Yy^2, Zz^2 <D Out[15]= 2 Xx + 2 Yy + 2 Zz In[16]:= SetCoordinates @ Cartesian @ x, y, z DD Out[16]= Cartesian @ x, y, z D In[17]:= Coordinates @D Out[17]= 8 x, y, z < A vector can be represented by a 3-element list : In[18]:= a = 8 ax @ x, y, z D , ay @ x, y, z D...
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This note was uploaded on 04/25/2010 for the course PHYS 5900 taught by Professor Lin during the Fall '09 term at East Carolina University .

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class14 - PHYS 5900 Class 14 (9/25/2008Fri) Zi-Wei Lin...

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