Analysis of Static Friction in Pure Rolling of a Disk

DOI: https://doi.org/10.62517/jnse.202617413

Author(s)

Ling Liu

Affiliation(s)

College of Intelligent Manufacturing, Jingchu University of Technology, Jingmen, Hubei, China

Abstract

Determining the size and direction of the static friction force at the contact point is still one of the more difficult problems to teach in theoretical mechanics, especially for a disk rolling without slipping on a rough inclined plane. To solve the above problem, we start with the differential equations of planar rigid-body motion and impose the kinematic constraint of no-slip rolling for the disk. Thus, an expression for the static friction force under a combined horizontal force and a couple load on the disk has been obtained. Next, we will study how some main factors influence the direction of friction, such as the size and position of the applied load, the slope of the plane, and the radius of gyration of the disk. The results indicate that pure rolling can be decomposed into two fundamental motion forms: translation of the center of mass and rotation about the center of mass. The role of static friction on these two motion forms is always reciprocal: in pure rolling induced by rotational action, static friction opposes rotation while promoting translation, with its direction being the same as the rolling direction; in pure rolling induced by translational action, static friction opposes translation while promoting rotation, with its direction being opposite to the rolling direction. For complex cases where both actions are present simultaneously, the direction of the resultant static friction can be determined by simplifying all active forces to the center of mass and superposing the individual components.

Keywords

Disk; Pure Rolling; Static Friction; Plane Motion of Rigid Body; Direction Criterion; Theoretical Mechanics Teaching.

References

[1] Xing G Z, Gao C Y. The role of static friction in pure rolling of a disk (rigid body). Journal of Liaoning Institute of Technology, 1994(4): 81-82. [2] Yao G H. Intuitive method for determining the direction of static friction and the work done by friction in rolling of a rigid body. College Physics, 1985(2): 23-24+49. [3] Jiang F, Zhao D. Determination of the direction of static friction on a purely rolling circular wheel. Mechanics in Engineering, 2011, 33(1): 86-88. [4] Liu D W, Niu F L. Determination of the direction of static friction on a purely rolling wheel. Journal of Gansu Lianhe University (Natural Sciences), 2008, 22(6): 51-53+104. [5] Zhen Z. Discussion on the direction of static friction on a body in pure rolling under different external forces. Physics Bulletin, 2011(9): 87-89. [6] Li H L, Guo Z J. The direction of static friction on a rigid body in pure rolling. Journal of Shanxi Normal University (Natural Science Edition), 1998(1): 43-45. [7] Chang J R, Jin J F, Song L N. Determination of the direction of static friction in pure rolling of a rigid body and its influencing factors. Journal of Jiangxi University of Science and Technology, 2009, 30(6): 42-44. [8] Hu J, Lu T S, Wu Q, et al. Analysis and discussion on the direction of friction and the work done by friction in pure rolling of a homogeneous rigid body. Physics and Engineering, 2019, 29(S1): 114. [9] Pang Y H, Hu Y J, Geng L L. Determination of the direction of static friction on a wheel. Physics Teacher, 2012, 33(11): 51. [10] Theoretical Mechanics Teaching and Research Section, Harbin Institute of Technology. Theoretical Mechanics (9th Edition, Volume I). Beijing: Higher Education Press, 2023: 216. [11] Cui M Z. Rolling resistance is not friction. Proceedings of the 11th National Conference on Tribology. Lanzhou: State Key Laboratory of Solid Lubrication, Lanzhou Institute of Chemical Physics, Chinese Academy of Sciences; Tribology Branch of Chinese Mechanical Engineering Society, 2013: 6.