Pure Circular Motion with Non-Angular Variables in One-Dimensional Motion Physics Problems

Daniel Blessner

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Conference: 2022 Spring ASEE Middle Atlantic Section Conference
Location: Newark, New Jersey
Publication Date: April 22, 2022
Start Date: April 22, 2022
End Date: April 23, 2022
Page Count: 16
DOI: 10.18260/1-2--40064
Permanent URL: https://peer.asee.org/40064
Download Count: 413

Paper Authors

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Daniel Blessner Pennsylvania State University, Wilkes-Barre Campus

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I'm a faculty member at the Penn State Wilkes Barre campus. My undergraduate degree is in civil engineering and a masters degree in chemical engineering. I I have been teaching math and physics for 27 years.

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Abstract

Pure Circular Motion with Non-Angular Variables in One-Dimensional Motion Physics Problems In standard algebra and calculus-based college physics textbooks, chapter two normally introduces one-dimensional motion for a particle with the following equations v = dx/dt and a = dv/dt. The kinematic equations with constant acceleration are then derived for motion along the x-axis. Then a second section is introduced for motion along the y-axis for freely falling objects in which the constant acceleration is ay = g = 9.81 m/s2. This pedagogical paper suggests that a third section be included for a particle travelling in pure circular motion. With the introduction of one new formula for Radial Acceleration, the standard kinematic equations can be used. The standard kinematic equations will be defined with tangential velocity and acceleration. This paper proposes to model particles travelling along the circumference of a circle with radius R relating displacement x as multiples of n Straight-Line segments of length C = 2πR along a horizontal axis with markings of 0C, 1C, 2C, 3C etc. Once the formula for radial acceleration is derived, the following axes will also be defined. The positive x and positive Tangential Velocity axes. For these problems we simplify things and will consider only counterclockwise motion. This direction will be considered positive. In addition, a moving particle will not come to rest and reverse direction. The positive x-axis can then be related to the positive circumference axis through the equation x = nC in which n is the number of circumference lengths travelled by the particle or the number of times the particle travelled around its circular path. Lastly, the tangential acceleration axis which will have both positive and negative acceleration values and perpendicular to that axis, the positive radial acceleration axis will be introduced. The positive radial acceleration axis will point to the center of the circle. Both uniform and non-uniform circular motion will be discussed. The difference in these pure circular motion problems versus one-dimensional motion along the x or y axes is that the resultant acceleration vector will have two components. In addition, an angle will need to be calculated between these two acceleration components. There will be no formulas that contain standard rotational variables such as ω and α. Also, a new notation for the introduction of time will be introduced. For example, with an initial velocity of 1 m/s and a given tangential acceleration how long will it take to reach a final velocity of 4m/s? The notation will be as follows (1m/s, 0s) and (4m/s,?s) This alternative method eliminates the need for plotting variables in a two-dimensional graph with time on the horizontal axis. This third introduced section of pure simple circular motion expressed as straight-line motion with the particle travelling along the positive n-axis can greatly help students visualize the more difficult general curvilinear motion seen in their sophomore year dynamics course. Except for the derivation of radial acceleration this paper will contain the full written section needed to be included in a standard Physics textbook with the appropriate number of fully worked example problems.

Citation
Format

Blessner, D. (2022, April), Pure Circular Motion with Non-Angular Variables in One-Dimensional Motion Physics Problems Paper presented at 2022 Spring ASEE Middle Atlantic Section Conference, Newark, New Jersey. 10.18260/1-2--40064

TY - CPAPER
AB - Pure Circular Motion with Non-Angular Variables in One-Dimensional Motion Physics Problems
In standard algebra and calculus-based college physics textbooks, chapter two normally introduces one-dimensional motion for a particle with the following equations v = dx/dt and a = dv/dt. The kinematic equations with constant acceleration are then derived for motion along the x-axis. Then a second section is introduced for motion along the y-axis for freely falling objects in which the constant acceleration is ay = g = 9.81 m/s2. This pedagogical paper suggests that a third section be included for a particle travelling in pure circular motion. With the introduction of one new formula for Radial Acceleration, the standard kinematic equations can be used. The standard kinematic equations will be defined with tangential velocity and acceleration.
This paper proposes to model particles travelling along the circumference of a circle with radius R relating displacement x as multiples of n Straight-Line segments of length C = 2πR along a horizontal axis with markings of 0C, 1C, 2C, 3C etc. Once the formula for radial acceleration is derived, the following axes will also be defined. The positive x and positive Tangential Velocity axes. For these problems we simplify things and will consider only counterclockwise motion. This direction will be considered positive. In addition, a moving particle will not come to rest and reverse direction. The positive x-axis can then be related to the positive circumference axis through the equation x = nC in which n is the number of circumference lengths travelled by the particle or the number of times the particle travelled around its circular path. Lastly, the tangential acceleration axis which will have both positive and negative acceleration values and perpendicular to that axis, the positive radial acceleration axis will be introduced. The positive radial acceleration axis will point to the center of the circle.
Both uniform and non-uniform circular motion will be discussed. The difference in these pure circular motion problems versus one-dimensional motion along the x or y axes is that the resultant acceleration vector will have two components.
In addition, an angle will need to be calculated between these two acceleration components. There will be no formulas that contain standard rotational variables such as ω and α.
Also, a new notation for the introduction of time will be introduced. For example, with an initial velocity of 1 m/s and a given tangential acceleration how long will it take to reach a final velocity of 4m/s? The notation will be as follows (1m/s, 0s) and (4m/s,?s)
This alternative method eliminates the need for plotting variables in a two-dimensional graph with time on the horizontal axis.
This third introduced section of pure simple circular motion expressed as straight-line motion with the particle travelling along the positive n-axis can greatly help students visualize the more difficult general curvilinear motion seen in their sophomore year dynamics course. Except for the derivation of radial acceleration this paper will contain the full written section needed to be included in a standard Physics textbook with the appropriate number of fully worked example problems.

AU - Daniel Blessner
CY - Newark, New Jersey
DA - 2022/04/22
PB - ASEE Conferences
TI - Pure Circular Motion with Non-Angular Variables in One-Dimensional Motion Physics Problems
UR - https://peer.asee.org/40064
DO - 10.18260/1-2--40064
ER -