POSSIBLE SOLUTION(S)

Write up of first situation - Problems 1-6:

We first define the variables:
radius = constant radius of circle
length = constant length of dragged stick
theta = the angle (we presume the disk rotates at 1 radian/sec)
xwheel(theta) = the horizontal distance of pin from the vertical radius
through the center of the rolling circle.
ywheel(theta) = the vertical distance of pin from the x-axis
xstick(theta) = the horizontal distance from the pin to the dragging
tip of the stick

Input := 

radius = 10; length = 40;

General::spell1: 
   Possible spelling error: new symbol name "length"
     is similar to existing symbol "Length".

We start the disk rolling at origin (0, 0) and the stick laying down l = 40 units to the left of the origin, i.e. the dragging tip of the stick is at (-40, 0).

Input := 

xwheel[theta_]= radius*Sin[theta];

Input := 

ywheel[theta_]= radius*(1 - Cos[theta]);

General::spell1: 
   Possible spelling error: new symbol name "ywheel"
     is similar to existing symbol "xwheel".

Input := 

xstick[theta_]= Sqrt[length^2 - ywheel[theta]^2];

xroll(theta) = the distance the circle has rolled along the horizontal
when it rotates through angle theta.

Input := 

xroll[theta_]= radius*theta;

xtotal(theta) = the actual x-coordinate of the dragging tip of the stick.

We obtain xtotal(theta) by subtracting xwheel(theta) (the horizontal distance of pin from the vertical radius through the center of the rolling circle) and xstick(theta) (the the horizontal distance from the pin to the dragging tip of the stick) from xroll(theta) (the the distance the circle has rolled along the horizontal when it rotates through angle theta).

Input := 

xtotal[theta_] = -xwheel[theta] - xstick[theta] + xroll[theta];

Here is a plot of the horizontal displacement of the dragging tip of the stick.

Input := 

Plot[xtotal[theta],{theta,0,4Pi}];

We compute and plot the velocity of the horizontal displacement of the dragging tip of the stick.

Input := 

vxtotal[theta_]=xtotal'[theta];
	Plot[vxtotal[theta],{theta,0,4Pi}];

General::spell1: 
   Possible spelling error: new symbol name "vxtotal"
     is similar to existing symbol "xtotal".

It appears that the dragging tip of the stick comes to a complete stop because the minimum value of the velocity looks to be 0.

We confirm this below by determining where the derivative of velocity, acceleration, is 0. Indeed we see that the tip of the stick is at rest at
theta = 2 Pi.

Input := 

axtotal[theta_] = vxtotal'[theta];
	Plot[axtotal[theta],{theta,0,4Pi}];

General::spell: 
   Possible spelling error: new symbol name "axtotal"
     is similar to existing symbols {vxtotal, xtotal}.

Input := 

solmin = FindRoot[axtotal[theta] == 0,{theta,6.4}]

Output =

{theta -> 6.28319}

Input := 

2 Pi//N

Output =

6.28319

Input := 

vxtotal[theta]/.solmin[[1]]

Output =

0.

Now as to the maximum velocity we find that this is 21.3541 cm/sec and this occurs at theta = 2.70334 when the wheel has not yet finished a half a revolution, i.e,. the maximum velocity DOES NOT occur when the pin is at the top of the wheel, while the minimum velocity DOES occur when the pin is at the bottom of the wheel.

Input := 

solmax = FindRoot[axtotal[theta] == 0,{theta,2}]

Output =

{theta -> 2.70334}

Input := 

vxtotal[theta]/.solmax[[1]]

Output =

21.3541

We plot the displacement, velocity, and acceleration of the function xtotal(theta), the location of the dragging stick contact.

Input := 

Plot[{xtotal[theta], vxtotal[theta],axtotal[theta]},
		{theta,0,4 Pi},
	  PlotStyle-> 
	  {Thickness[.005], Thickness[.01],Thickness[.015]}]

Output =

-Graphics-

The thickest curve is acceleration, the middle thickness curve is velocity, and the thinnest curve is position. We see the usual confirmations of one function's derivative being zero at its relative maxima and minima. We see some other interesting things here.

From the graph of the acceleration we note that acceleration peaks at 1.46974 sec to a value of 12.8564 cm/sec^2, again we have an asymmetry with respect to time, i.e. things do not happen at equal intervals in the period of the rolling wheel [0, 2 Pi] .

Input := 

jxtotal[theta_] = axtotal'[theta];

General::spell: 
   Possible spelling error: new symbol name "jxtotal"
     is similar to existing symbols 
    {axtotal, vxtotal, xtotal}.

Input := 

solamax = FindRoot[jxtotal[theta] == 0,{theta,1.5}]

General::spell1: 
   Possible spelling error: new symbol name "solamax"
     is similar to existing symbol "solmax".

Output =

{theta -> 1.46974}

Input := 

axtotal[theta]/.solamax[[1]]

Output =

12.8564

The motion would appear to be stop and go, with an initial fast go (due to increasing acceleration in the interval [0, 1.46974]) and then decelerates.

We animate the motion of the stick and the disk.

Input := 

Do[Show[Graphics[{PointSize[.02],Point[{xtotal[theta],0}],
		Point[{radius theta - xwheel[theta],ywheel[theta]}],
		Circle[{radius theta,radius},radius],
		Line[{{-40,0},{8 Pi radius + radius,0}}],
		Line[{{xtotal[theta],0},
		  {radius theta - xwheel[theta],ywheel[theta]}}]}],
		PlotRange->{{-length - 10,8 Pi radius+ radius},
					{-5,2 radius + 5}},
			AspectRatio->Automatic],
			{theta,0, 8 Pi, Pi/8}]

Write up of second situation - Problems 11-20:

We consider the following situation where the length of the stick is equal to the diameter (l = 2 r). First we remove variable from the previous problem and redefine all the variables and functions.

Input := 

Remove[radius,length,xwheel,ywheel,xstick,xroll,vxtotal,axtotal,jxtotal]

Input := 

radius =20; length = 40;

General::spell1: 
   Possible spelling error: new symbol name "length"
     is similar to existing symbol "Length".

We start the disk rolling at origin (0, 0) and the stick laying down l = 40 units to the left of the origin, i.e. the dragging tip of the stick is at (-40, 0).

Input := 

xwheel[theta_]= radius*Sin[theta];

Input := 

ywheel[theta_]= radius*(1 - Cos[theta]);

General::spell1: 
   Possible spelling error: new symbol name "ywheel"
     is similar to existing symbol "xwheel".

Input := 

xstick[theta_]= Sqrt[length^2 - radius^2(1-Cos[theta])^2];

Input := 

xroll[theta_]= radius*theta;

Input := 

xtotal[theta_] = -xwheel[theta] - xstick[theta] + xroll[theta];

Here is a plot of the horizontal displacement of the dragging tip of the stick.

Input := 

Plot[xtotal[theta],{theta,0,4Pi}];

We compute the velocity and acceleration of the horizontal displacement of the dragging tip of the stick.

Input := 

vxtotal[theta_] = xtotal'[theta]; axtotal[theta_] = vxtotal'[theta];

General::spell1: 
   Possible spelling error: new symbol name "vxtotal"
     is similar to existing symbol "xtotal".

General::spell: 
   Possible spelling error: new symbol name "axtotal"
     is similar to existing symbols {vxtotal, xtotal}.

We compute and plot the velocity of the horizontal displacement of the dragging tip of the stick.

Input := 

vxtotal[theta_]=xtotal'[theta];
	Plot[vxtotal[theta],{theta,0,4Pi}];

It appears that the dragging tip of the stick comes to a complete stop because the minimum value of the velocity looks to be 0.

We confirm this below by determining where the derivative of velocity, acceleration, is 0. Indeed we see that the tip of the stick is at rest at
theta = 2 Pi.

Input := 

axtotal[theta_] = vxtotal'[theta];
	Plot[axtotal[theta],{theta,0,4Pi}];

Input := 

solmin = FindRoot[axtotal[theta] == 0,{theta,6.4}]

Output =

{theta -> 6.28319}

Input := 

2 Pi//N

Output =

6.28319

Input := 

vxtotal[theta]/.solmin[[1]]

Output =

0.

Now as to the maximum velocity we find that this is 31.547 cm/sec and this occurs at theta = Pi/2 when the wheel has finished a half a revolution, i.e,. the maximum velocity occurs when the pin is at the top of the wheel and the minimum velocity occurs when the pin is at the bottom of the wheel. We note that the maximum velocity is greater in this case and occurs just when the pin is just at the top path, as the stick is being "pulled upright" most rapidly.

Input := 

vxtotal[Pi/2]//N

Output =

31.547

We plot the displacement, velocity, and acceleration of the function xtotal(theta), the location of the dragging stick contact.

Input := 

Plot[{xtotal[theta], vxtotal[theta],axtotal[theta]},
		{theta,0,4 Pi},
	  PlotStyle-> 
	  {Thickness[.005], Thickness[.01],Thickness[.015]}]

Output =

-Graphics-

The thickest curve is acceleration, the middle thickness curve is velocity, and the thinnest curve is position. We see sharp cusp on the position function at theta = Pi/2 (when the pin is at the top of the wheel and the stick is fully vertical). This gives rise to a discontinuity in the derivative, indeed, there is no derivative at these point. When we take the second derivatives there appears to be a second derivative at these points of discontinuity in the first derivative. But the second derivatives are both approaching 0 from the left and from the right and so we seem to have a derivative at these points, but the second derivative does not exist at these points.

From the graph of the acceleration we note that acceleration peaks at 1.66536 sec to a value of 35.5782 cm/sec^2, (greater than in the shorter stick version above) again we have an asymmetry with respect to time, i.e. things do not happen at equal intervals in the period of the rolling wheel [0, 2 Pi] .

Input := 

jxtotal[theta_] = axtotal'[theta];

General::spell: 
   Possible spelling error: new symbol name "jxtotal"
     is similar to existing symbols 
    {axtotal, vxtotal, xtotal}.

Input := 

solamax = FindRoot[jxtotal[theta] == 0,{theta,1.5}]

Output =

{theta -> 1.66536}

Input := 

axtotal[theta]/.solamax[[1]]

Output =

35.5782

The motion would appear to be stop and go, with an initial fast go (due to increasing acceleration in the interval [0, 1.66536]) and then decelerates.

We animate the motion of the stick and the disk.

Input := 

Do[Show[Graphics[{PointSize[.02],Point[{xtotal[theta],0}],
		Point[{radius theta - xwheel[theta],ywheel[theta]}],
		Circle[{radius theta,radius},radius],
		Line[{{-40,0},{8 Pi radius + radius,0}}],
		Line[{{xtotal[theta],0},
		  {radius theta - xwheel[theta],ywheel[theta]}}]}],
		PlotRange->{{-length - 10,8 Pi radius+ radius},
						{-5,2 radius + 5}},
			AspectRatio->Automatic],
			{theta,0, 8 Pi, Pi/8}]

POSSIBLE SOLUTION(S)

Write up of first situation - Problems 1-6:

We start the disk rolling at origin (0, 0) and the stick laying down l = 40 units to the left of the origin, i.e. the dragging tip of the stick is at (-40, 0).

xroll(theta) = the distance the circle has rolled along the horizontal when it rotates through angle theta.

Here is a plot of the horizontal displacement of the dragging tip of the stick.

We compute and plot the velocity of the horizontal displacement of the dragging tip of the stick.

It appears that the dragging tip of the stick comes to a complete stop because the minimum value of the velocity looks to be 0. We confirm this below by determining where the derivative of velocity, acceleration, is 0. Indeed we see that the tip of the stick is at rest at theta = 2 Pi.

We plot the displacement, velocity, and acceleration of the function xtotal(theta), the location of the dragging stick contact.

The thickest curve is acceleration, the middle thickness curve is velocity, and the thinnest curve is position. We see the usual confirmations of one function's derivative being zero at its relative maxima and minima. We see some other interesting things here.

From the graph of the acceleration we note that acceleration peaks at 1.46974 sec to a value of 12.8564 cm/sec^2, again we have an asymmetry with respect to time, i.e. things do not happen at equal intervals in the period of the rolling wheel [0, 2 Pi] .

The motion would appear to be stop and go, with an initial fast go (due to increasing acceleration in the interval [0, 1.46974]) and then decelerates.

We animate the motion of the stick and the disk.

Write up of second situation - Problems 11-20:

We consider the following situation where the length of the stick is equal to the diameter (l = 2 r). First we remove variable from the previous problem and redefine all the variables and functions.

We start the disk rolling at origin (0, 0) and the stick laying down l = 40 units to the left of the origin, i.e. the dragging tip of the stick is at (-40, 0).

Here is a plot of the horizontal displacement of the dragging tip of the stick.

We compute the velocity and acceleration of the horizontal displacement of the dragging tip of the stick.

We compute and plot the velocity of the horizontal displacement of the dragging tip of the stick.

It appears that the dragging tip of the stick comes to a complete stop because the minimum value of the velocity looks to be 0. We confirm this below by determining where the derivative of velocity, acceleration, is 0. Indeed we see that the tip of the stick is at rest at theta = 2 Pi.

We plot the displacement, velocity, and acceleration of the function xtotal(theta), the location of the dragging stick contact.

The motion would appear to be stop and go, with an initial fast go (due to increasing acceleration in the interval [0, 1.66536]) and then decelerates.

We animate the motion of the stick and the disk.

xroll(theta) = the distance the circle has rolled along the horizontal
when it rotates through angle theta.

It appears that the dragging tip of the stick comes to a complete stop because the minimum value of the velocity looks to be 0.

We confirm this below by determining where the derivative of velocity, acceleration, is 0. Indeed we see that the tip of the stick is at rest at
theta = 2 Pi.

It appears that the dragging tip of the stick comes to a complete stop because the minimum value of the velocity looks to be 0.

We confirm this below by determining where the derivative of velocity, acceleration, is 0. Indeed we see that the tip of the stick is at rest at
theta = 2 Pi.