San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Dynamic Appearance of a Rotating Object

This is an investigation of the appearance of a rotating object to a stream of particle such as electrons.

Particle Stream Parallel to the Axis of Rotation

Consider a simple fan. If the fan is motionless and dark particles are streamed at its circular enclosure the result would be a clear-cut silhouette.

If the fan is slowly turning the silhouette would be replaced by circle lightened by the white area spread evenly over the circular enclosure of the fan.

If the fan is turning faster it will prevent passage of particles over a greater portion of the enclosure, as depicted below.

This will give light to a lighter image such as:

The Determination of the Degree of Opaqueness
of the Image of a Spinning Fan

Let ω be the angular rate of rotation of the fan. Let D be the effective thickness of the fan parallel to the axis of rotation and v be the velocity of the stream of particles. The time that the particles are subject to being swept away by the fan is Δt=D/v. The angle that the fan turns in that among of time is

Δθ = (D/v)ω

and hence the angle covered by the fan is

θ0 + (D/v)ω

where θ0 is the angular width of the fan blade. The shade of the spinning image is then a weighted average of the shade of the fan and the shade of the background; i.e.,

Shade of image = γ(Shade of fan) + (1-γ)(Shade of background)

where γ = (θ0 + (D/v)ω)/(2π) provided that (θ0 + (D/v)ω) is less than or equal to 2π, otherwise γ=1.

Particle Stream Perpendicular to the Axis of Rotation

Consider an annular ring such as:

If the ring is fixed in the vertical position and a stream of particles impinges upon it a silhouette is created as below.

If the ring is rotating about a horizontal diameter smeared image is created by the particles that pass through.

Visually a flipping ring would create an image more like the one shown below in which there is a reversal of black and white compared to the previous image.

(To be continued.)

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