Rolling without slipping is the motion of a round object that rotates and moves forward without sliding at the contact point. It matters because it connects straight-line motion, rotation, friction, and energy in one common situation. Wheels on cars, bicycle tires, bowling balls, and cylinders rolling down ramps all use the same basic physics.
The key condition is that the center of mass speed equals the angular speed times the radius.
Key Facts
- Rolling condition: v = ωr
- The contact point is instantaneously at rest relative to the ground when there is no slipping.
- Total kinetic energy: K = 1/2 mv^2 + 1/2 Iω^2
- For rolling down a height h: mgh = 1/2 mv^2 + 1/2 Iω^2
- Acceleration down an incline: a = g sinθ / (1 + I/(mr^2))
- Objects with smaller I/(mr^2) accelerate faster down the same ramp.
Vocabulary
- Rolling without slipping
- Motion in which a round object rotates and translates so that the point touching the surface does not slide.
- Center of mass
- The average position of an object's mass, which moves as if the object's total mass were concentrated there.
- Angular speed
- The rate at which an object rotates, usually measured in radians per second.
- Moment of inertia
- A measure of how strongly an object resists changes in rotational motion.
- Static friction
- The friction force between surfaces that are not sliding past each other.
Common Mistakes to Avoid
- Using v = ω instead of v = ωr is wrong because angular speed and linear speed have different units and must be related by the radius.
- Assuming the contact point is moving forward at speed v is wrong because in pure rolling the contact point is instantaneously at rest relative to the surface.
- Ignoring rotational kinetic energy is wrong because a rolling object stores energy in both translation and rotation.
- Thinking friction always removes mechanical energy is wrong because static friction can provide torque without doing work on an ideal rolling object.
Practice Questions
- 1 A wheel of radius 0.30 m rolls without slipping with angular speed 12 rad/s. What is the speed of its center of mass?
- 2 A solid cylinder with I = 1/2 mr^2 rolls from rest down a vertical height of 2.0 m without slipping. Use g = 9.8 m/s^2 to find its final center of mass speed.
- 3 A solid cylinder and a hoop have the same mass and radius and are released from rest at the top of the same ramp. Which reaches the bottom first, and why?