Rolling Motion - University Physics Volume 1 | OpenStax

Learning Objectives

• Understand the physics of rolling motion without slipping
• Relate linear and angular variables in rolling motion
• Calculate linear and angular accelerations with and without slipping
• Determine static friction force in rolling motion
• Apply energy conservation in analyzing rolling motion

Introduction to Rolling Motion

• Rolling motion is a combination of rotational and translational motion.
• Key examples include:
  • Wheels on cars, airplanes, and robotic explorers.
• Understanding forces and torques in rolling motion is crucial.

Rolling Motion without Slipping

• Observed since the invention of the wheel.
• Static friction keeps cars' tires in contact with the road during rolling.
  • Without slipping, the bottom of the wheel is at rest relative to the ground.
• Key Points:
  • Linear velocity and acceleration can be derived from angular variables.
  • v_CM = ωR a_CM = αR
  • Distance traveled: d_CM = θR

Example: Rolling Down an Inclined Plane

• Analyze a solid cylinder rolling down without slipping.
• Forces involved: Normal force, gravitational force, static friction.
• Linear acceleration expressed using moment of inertia.
  • Formula: a_CM = \frac{mgsinθ}{m + \frac{I_{CM}}{R^2}}
• Static friction must be sufficient to prevent slipping.
• Relation between static friction and incline angle: s ≥ \frac{1}{3}tanθ

Rolling Motion with Slipping

• Kinetic friction is involved when slipping occurs.
• Relationships between linear and angular variables change.
  • Key Relationships:
  • v_CM ≠ ωR a_CM ≠ αR

Example: Rolling Down an Incline with Slipping

• Solve using kinetic friction instead of static friction.
• Formula for acceleration: a_CM = g(sinθ - μ_kcosθ)
• Angular acceleration depends on kinetic friction and incline: α = \frac{2μ_kgcosθ}{R}

Conservation of Mechanical Energy in Rolling Motion

• Total mechanical energy includes translational and rotational kinetic energy, and potential energy: E_T = \frac{1}{2}mv_{CM}^2 + \frac{1}{2}I_{CM}ω^2 + mgh
• Energy conserved if no nonconservative forces (e.g., slipping, air resistance).
• Example: Curiosity on Mars
  • Wheel rolling without slipping.
  • Use energy conservation to find velocity:
  v_CM = \sqrt{gh}

Summary

• Rolling motion combines translation and rotation.
• Static friction is central to rolling without slipping.
• Energy conservation useful in predicting motion outcomes.
• Understanding the physics helps in analyzing real-world applications, such as vehicle dynamics and planetary exploration.