The Physics
Hypertextbook
Opus in profectus

# Rolling

## Summary

• Symbols used in this section  r = radius in the general sense (distance from the center or axis of rotation) R = the outer radius of a round object (often just called the radius of the object) vcm = translational speed of the center of mass ω = rotational or angular speed
• Rolling is a combination of translational and rotational motion.
• When an object experiences pure translational motion, all of its points…
• move with the same velocity as the center of mass; that is…
• in the same direction
• with the same speed (v = vcm)
• move in a straight line in the absence of a net external force
• When an object experiences pure rotational motion about its center of mass, all of its points…
• move at right angles to the radius in a plane perpendicular to the axis of rotation, thus…
• points on opposite sides of the axis of rotation move in opposite directions
• move with a speed proportional to radius (v = rω), thus…
• the center of mass does not move (since r = 0 there)
• points on the outer radius move with speed v = Rω
• move in a circle centered on the axis of rotation
• When an object experiences rolling motion
• the point of the object in contact with the surface…
• is instantaneously at rest
• is the instantaneous axis of rotation
• the center of mass of the object…
• moves with speed vcm = Rω
• moves in a straight line in the absence of a net external force
• the point fathest from the point of contact…
• moves with twice the speed of the center of mass v = 2vcm = 2Rω
• Rolling and Slipping
• rolling without slipping
• vcm = Rω
• slipping
• and rolling forward
• vcm < Rω
• accelerating on ice or mud
• "burnout" or "burn rubber" while driving
• "top spin" in billiards (a.k.a. "top" or "follow")
• vcm > Rω
• decelerating on ice or mud
• and rolling backward
• vcm > 0 and ω < 0
• "back spin" in billiards (a.k.a. "bottom" or "draw")
• pure translation
• vcm ≠ 0 and ω = 0
• "wheel lock" while driving
• "slide" in billiards
• pure rotation
• vcm = 0 and ω ≠ 0
• stuck in mud or snow while driving
• The path of a point on a rolling object is a cycloid (or a trochoid).
• The cycloid generated by a point on an object rolling over the +x axis is described by the following parametric equations…  rolling translation rotation = + x = vcmt + r cos(θ − ωt) y = r + r sin(θ − ωt)
where…  r, θ = cylindrical coordinates of the point R = outer radius vcm = translational speed of the center of mass ω = rotational or angular speed t = time (the parameter of the parametric equation)
• Types
• A basic cycloid…
• is traced out by…
• points on the surface of a generating circle that is…
• rolling without slipping
• over a straight line
• has cusps (points with two tangents)
• A cycloid is curtate (or contracted) if…
• it is traced out by…
• points inside a generating circle (r < R) that is rolling without slipping or
• points on the surface of the generating circle that is slipping while rolling with vcm > Rω
• does not have cusps or loops
• A cycloid is prolate (or extended) if…
• it is traced out by…
• points outside a generating circle (r > R) that is rolling without slipping or
• points on the surface of the generating circle that is slipping while rolling with vcm < Rω
• it has loops
• A cycloid formed by rolling a generating circle on another circle is called…
• an epicycloid if the generating circle rolls on the outside of the other circle
• a hypocycloid if the generating circle rolls on the inside of the other circle