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)
v_cm	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 and
    - with the same speed (v = v_cm)
  - 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) and
    - 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 v_cm = 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 = 2v_cm = 2Rω
Rolling and Slipping
- rolling without slipping
- slipping
  - and rolling forward
    - v_cm < Rω
      - accelerating on ice or mud
      - "burnout" or "burn rubber" while driving
      - "top spin" in billiards (a.k.a. "top" or "follow")
    - v_cm > Rω
      - decelerating on ice or mud
  - and rolling backward
    - v_cm > 0 and ω < 0
      - "back spin" in billiards (a.k.a. "bottom" or "draw")
- pure translation
- pure rotation

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	=	v_cmt	+	r cos (θ − ωt)
y	=	R	+	r sin (θ − ωt)

where …

r, θ	cylindrical coordinates of the point
R	outer radius
v_cm	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 v_cm > 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 v_cm < 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

The Physics Hypertextbook

No condition is permanent.