A video of this demonstration is available at this link.

You can illustrate the variables of rotational motion with this bicycle wheel.

The demonstrations in section 08 -- Kinematics, allow you to show straight-line motion at constant velocity, accelerated straight-line motion and linear motion in two dimensions. With their aid, you can illustrate the concepts of position, (linear) velocity and (linear) acceleration, and derive the equations of motion in a straight line and in a plane. (Two of the demonstrations in section 04 -- Measurement, allow you to show the Cartesian coordinate system and vectors, which are essential to any discussion of the aforementioned concepts.) In similar fashion, this demonstration allows you to show rotational motion, that is, motion in which an object rotates about a particular axis. If we take the

x-axis as the horizontal line that passes through the center of the wheel, and they-axis as the vertical line passing through the center of the wheel, then the axle of the wheel points along thez-axis. The wheel, of course, rotates about thez-axis. A white stripe on the rim serves as a reference for the position of the wheel.Whereas in linear motion, we denote a position along the

x- ory-axis (or both) withxandycoordinates, in rotational motion, we describe position in terms of anangulardisplacement with respect to a reference position, say thex-axis. By convention, the positive direction is counterclockwise, and the angle,θ, equalss/r, whereris the radius of the circle along which the point on the object moves, andsis the length of the arc from zero to its present position. So, taking the positivex-axis as the zero reference, the white stripe on the wheel in the photograph above is at an angular position of about 5.5 radians (≈ 315°).In what follows, we consider the wheel as a

rigid body, that is, we assume that all points on the wheel maintain the same spatial relationship to each other as the wheel rotates.If the the wheel is rotating at constant speed, its change in position with respect to time is a change in angle with respect to time, Δ

θ/Δt. This is itsaverage angular speed. Theinstantaneous angular speedis the same ratio in the limit as Δtgoes to zero, usually denoted as ω; ω =dθ/dt. This is also the magnitude ofω(=d/θdt), theangular velocity, which points along the axle of the wheel. If the wheel is rotating in the counterclockwise direction, then according to the right-hand rule,ωpoints outwards, away from the table. The units of ω are typically rad/s. Depending on context, one also sees rev/s or deg/s.If the speed of rotation of the wheel is not constant, then the wheel is accelerating, and ω is changing with time. As with angular speed, the

average angular accelerationis Δω/Δt, and, in the limit as Δtgoes to zero, theinstantaneous angular acceleration,α, equalsdω/dt. Its units are typically rad/s^{2}, but depending on context, you might also see rev/s^{2}or deg/s^{2}. This is also a scalar quantity that corresponds to the magnitude of a vector,α, which equalsd/ωdt. As for,ωpoints according to the right-hand rule. If we accelerate the wheel in our example in the counterclockwise direction, so that its speed is constantly increasing in that direction, thenαpoints outwards, away from the table.αOur earlier statement about rigid bodies means that both

ωandαare the same for all points on the wheel (or other rotating object).The scalar quantities

θ,ω(=dθ/dt) and α (=dω/dt=d^{2}θ/dt^{2}) are analogous to the scalar quantitiesx,v(=dx/dt) anda(=dv/dt=d^{2}x/dt^{2}), respectively, and the vector quantities,θ(=ωd/θdt) and(=αd/ωdt=d^{2}/θdt^{2}) are analogous to the vector quantitiesr,v(=dr/dt) anda(=dv/dt=d^{2}r/dt^{2}), respectively. This leads to a set of equations that are analogous to those for linear motion (given here in scalar form, assuming motion in thexdirection and thatx_{0}= 0):

Linear motion Rotational motionv=v_{0}+atω=ω_{0}+αtx= (1/2)(v_{0}+v)tθ=(1/2)(ω_{0}+ω)tx=v_{0}t+ (1/2)at^{2}θ=ω_{0}t+ (1/2)αt^{2}v^{2}=v_{0}^{2}+ 2axω^{2}=ω_{0}^{2}+ 2αθWe should also note that when the wheel is rotating at a constant speed, though all points on the wheel have the same angular speed,

ω, theirinstantaneous linear speeds, in the direction tangent to the circles in which they are rotating, are proportional to their distance from the axis of rotation. If we call this instantaneous linear speedv, for a point at a distancerfrom the axis of the wheel,v=ωr. Similarly, though all points have the same angular acceleration, theirinstantaneous linear accelerations(again tangential to the circle of motion) are proportional to their distance from the axis of rotation, ora=αr.

References:1) Resnick, Robert and Halliday, David.

Physics, Part One, Third Edition(New York: John Wiley and Sons, 1977), pp. 32, 215-223.