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
- move with the same velocity as the center of mass; that is…
- 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
- move at right angles to the radius in a plane perpendicular to the axis of rotation, thus…
- 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
- the point of the object in contact with the surface…
- When an object experiences pure translational motion, all of its points…
- 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
- vcm < Rω
- and rolling backward
- vcm > 0 and ω < 0
- "back spin" in billiards (a.k.a. "bottom" or "draw")
- vcm > 0 and ω < 0
- and rolling forward
- pure translation
- vcm ≠ 0 and ω = 0
- "wheel lock" while driving
- "slide" in billiards
- vcm ≠ 0 and ω = 0
- pure rotation
- vcm = 0 and ω ≠ 0
- stuck in mud or snow while driving
- vcm = 0 and ω ≠ 0
- rolling without slipping
- 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) 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)
- is traced out by…
- 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
- it is traced out by…
- 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
- it is traced out by…
- 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
- A basic cycloid…
- The cycloid generated by a point on an object rolling over the +x axis is described by the following parametric equations…