# 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
- with the same speed (
*v*=*v*)_{cm}

- 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*=*r*ω), thus…- the center of mass does not move (since
*r*= 0 there) - points on the outer radius move with speed
*v*=*R*ω

- the center of mass does not move (since
- 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
*v*=_{cm}*R*ω - moves in a straight line in the absence of a net external force

- moves with speed
- the point fathest from the point of contact…
- moves with twice the speed of the center of mass
*v*= 2*v*= 2_{cm}*R*ω

- moves with twice the speed of the center of mass

- 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
*v*=_{cm}*R*ω

- slipping
- and rolling forward
*v*<_{cm}*R*ω- accelerating on ice or mud
- "burnout" or "burn rubber" while driving
- "top spin" in billiards (a.k.a. "top" or "follow")

*v*>_{cm}*R*ω- decelerating on ice or mud

- and rolling backward
*v*> 0 and ω < 0_{cm}- "back spin" in billiards (a.k.a. "bottom" or "draw")

- and rolling forward
- pure translation
*v*≠ 0 and ω = 0_{cm}- "wheel lock" while driving
- "slide" in billiards

- pure rotation
*v*= 0 and ω ≠ 0_{cm}- stuck in mud or snow while driving

- 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*= *v*_{cm}t+ *r*cos(θ − ω*t*)*y*= *r*+ *r*sin(θ − ω*t*)*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)

- 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*v*>_{cm}*R*ω

- points
- 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*v*<_{cm}*R*ω

- points
- 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

- an epicycloid if the generating circle rolls on the

- A basic cycloid…

- The cycloid generated by a point on an object rolling over the +