On Rotating and Rolling

Judah L Schwartz, Apr 11, 2016

This is a series of applets drawn from the website mathMINDhabits [[url]sites.google.com/site/mathmindhabits[/url]] that is designed for teachers of mathematics who want to deepen their understanding of the mathematics they teach and that their students are expected to learn. The applets in this collection all focus on circular and/or rotational motion in different settings. In the first chapter the first two applets deal with a circle rolling around either the outside or the inside of a second circle. Many people find the phenomenon puzzling. The search for a resolution of the puzzle provides a good deal of insight into the nature of rotational motion. The next two applets deal with a rolling wheel with a pebble stuck on the outer rim of the wheel. The trajectory of the pebble is then displayed both in space and in time. The "Pebble stuck in a Wheel" applet displays the height of the pebble as a function of distance along the road and as a function of time. The Pebble stuck in a Wheel - revisited" displays these same trajectories in a single three dimensional [distance, height, time] space. The final applet in the first chapter, "Rolling Disk Illusion", clarifies the need to be clear about the nature of the points being followed in describing rotational motion. The second chapter applets "two lap average" and "two lap average - revisited" both explore the question of how average speed varies as a driver traverses a closed course during two successive laps. "A Run for Two" lets the user consider how the separation between two runners varies as a function of time as they run around a circular track, either in the same or in opposite directions. The applets in the third chapter deal with planetary motion. In "Two Planets & A Sun - A Digital Triptych" the motions of a sun and two of its planets are seen in each of their coordinate systems. Retrograde motion of planets becomes more readily understandable in such displays. The applet "Planetary Orbits in Two Frames" displays the motion of a sun-planet system as seen in the frame of the center-of-mass of the two bodies as well as the motion as seen from a vantage point in which the center-of-mass moves with constant velocity. The applets in the fourth chapter all deal with motion in a spiral. The first applet, "How a DVD Player Works" explores the motion of the rotating disk and the way the information on the disk is recorded and played back. The next two applets, "A Road Up A Mountain" and "Another Road Up A Mountain", both deal with the problem of possible paths of fixed slope up a mountain - in the first case the mountain is shaped like a cone, in the second case like an inverted parabolic bowl. A final applet in this chapter displays a zoo of classic spirals. The final chapter shows a classic paradox called "Aristotle's Wheel".

Table of Contents

  1. Rolling discs
    1. Disc rolling on a Disc
    2. Disc rolling inside a Disc
    3. Gear train
    4. Pebble stuck in a wheel -traditional view
    5. Pebble stuck in a wheel - space-time view
    6. Rolling Disk Illusion
  2. Around the track
    1. two lap average
    2. two lap average - revisited
    3. A Run for Two
    4. A Run for Two - revised
  3. Planetary motion
    1. Two Planets & A Sun - A Digital Triptych
    2. planetary orbits in two frames
  4. Spiraling in & out
    1. How a DVD player works
    2. a road up a mountain
    3. another road up a mountain
    4. A swirl of spirals
  5. Aristotle's Wheel
    1. Aristotle's Wheel

1.

Rolling discs

The first two applets in this chapter deal with a circle rolling around either the outside or the inside of a second circle. Many people find the phenomenon puzzling. The search for a resolution of the puzzle provides a good deal of insight into the nature of rotational motion. The next applet "Gear train" invites the exporation of the relationship between linear and angular velocity. The next two applets deal with a rolling wheel with a pebble stuck on the outer rim of the wheel. The trajectory of the pebble is then displayed both in space and in time. The "Pebble stuck in a Wheel" applet displays the height of the pebble as a function of distance along the road and as a function of time. The Pebble stuck in a Wheel - revisited" displays these same trajectories in a single three dimensional [distance, height, time] space. The final applet in this chapter, "Rolling Disk Illusion", clarifies the need to be clear about the nature of the points being followed in describing rotational motion.

  • 1. Disc rolling on a Disc
  • 2. Disc rolling inside a Disc
  • 3. Gear train
  • 4. Pebble stuck in a wheel -traditional view
  • 5. Pebble stuck in a wheel - space-time view
  • 6. Rolling Disk Illusion

1.1.

Disc rolling on a Disc

A GOLD disc rolls without slipping around the edge of a SILVER disc.[br][br]If the discs are the same size, how many revolutions does the GOLD disc make as it rolls around the SILVER disc exactly once?[br][br]You may find the result surprising? Can you explain it?[br][br]What happens if the discs are not the same size – if the radius of the GOLD is half that of the SILVER? twice? three times?[br][color=#ff0000][i][b][br]What questions could/would you ask of your students based on this applet?[/b][/i][/color]
– Judah L Schwartz

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

–

2.

Around the track

Two applets in this chapter - "two lap average" and "two lap average - revisited" both explore the question of how average speed varies as a driver traverses a closed course during two successive laps. A third applet, "A Run for Two", lets the user consider how the separation between two runners varies as a function of time as they run around a circular track, either in the same or in opposite directions.

  • 1. two lap average
  • 2. two lap average - revisited
  • 3. A Run for Two
  • 4. A Run for Two - revised

2.1.

two lap average

Andrei drives a 60 mile closed course in 1 hour.[br][br]How fast does he have to drive a second lap in order to average 120 mi/hr for the two laps?[br][br][size=85][b][see challenge below][/b][/size][br][br]
The applet makes clear that it is [i][b]not possible[/b][/i] for the two lap average to be equal to 120 mi/hr.[br][br]Can you prove this algebraically?
– Judah L Schwartz

2.2.

–

2.3.

–

2.4.

–

3.

Planetary motion

The applets in this chapter deal with planetary motion. In "Two Planets & A Sun - A Digital Triptych" the motions of a sun and two of its planets are seen in each of their coordinate systems. Retrograde motion of planets becomes more readily understandable in such displays. The applet "Planetary Orbits in Two Frames" displays the motion of a sun-planet system as seen in the frame of the center-of-mass of the two bodies as well as the motion as seen from a vantage point in which the center-of-mass moves with constant velocity.

  • 1. Two Planets & A Sun - A Digital Triptych
  • 2. planetary orbits in two frames

3.1.

Two Planets & A Sun - A Digital Triptych

The motions of a Sun and two planets as seen in each of their coordinate systems.[br][br]You can animate the motion by clicking on the lower left hand corner of the screen.[br][br]You can show the trajectories by checking the trajectories checkbox.[br][br]You can show the distances between the bodies by checking the distance checkbox.[br][br]Does this help you understand the retrograde motion of Mars?[br][br]Why does the triangle of distances look the same in all three coordinate systems?
– Judah L Schwartz

3.2.

–

4.

Spiraling in & out

The applets in this chapter all deal with motion in a spiral. The first applet, "How a DVD Player Works" explores the motion of the rotating disk and the way the information on the disk is recorded and played back. The next two applets, "A Road Up A Mountain" and "Another Road Up A Mountain", both deal with the problem of possible paths of fixed slope up a mountain - in the first case the mountain is shaped like a cone, in the second case like an inverted parabolic bowl. The final applet in this chapter displays a zoo of classic spirals.

  • 1. How a DVD player works
  • 2. a road up a mountain
  • 3. another road up a mountain
  • 4. A swirl of spirals

4.1.

How a DVD player works

Information is encoded on a DVD in the form of small uniformly spaced[br]pits on a long arithmetic spiral starting close to the center and spiraling [br]out to the outer edge of the disk.[br][br]A laser reading head moves from near the center along a radius[br]toward the outer edge.[br][br]If the disk turns at a constant angular velocity, pits near[br]the center would pass under the reading head more slowly than pits [br]near the outer edge. In order for the DVD to be read properly, the angular velocity[br]of the disk must vary with the position of the reading head.[br][br]The right hand panel shows the relationship among the relevant quantities.[br]If the radius of the first pit is [i][b][size=150]r[/size][/b][/i], that of the last pit is [i][b][size=150]R[/size][/b][/i], the [br]angular velocity of the turning disk is[size=150][i][b] f[/b][/i][/size], derive an expression for [br]rate at which angular velocity of the disk varies with the position[br]of the reading head.
– Judah L Schwartz

4.2.

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4.3.

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4.4.

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5.

Aristotle's Wheel

[i][b]A classic paradox ![/b][/i]

  • 1. Aristotle's Wheel

5.1.

Aristotle's Wheel

This animation seems to present a paradox. [br][br]A green dot is placed on the outer rim of a wheel.[br]A blue dot is placed on the inner radius of the wheel.[br][br][You can change the size of the inner radius by dragging the BLUE dot.][br][br]Run the animation - do you see the difficulty ?[br][br]Do you see a problem? [br][color=#ff0000][i][b][br]What questions could/would you ask your students based on this applet?[/b][/i][/color]
Adapted from an applet by John Golden
– Judah L Schwartz

Information