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The Marble moving on a grooved spiral

Question # 00178856
Subject: Engineering
Topic: General Mechanical Engineering
Due on: 02/21/2016
Posted On: 01/22/2016 03:36 PM

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On spirals that rotate
We consider the disk of diameter R0 = 1m shown in Fig. 3.

Figure 3: Marble moving on a grooved spiral
A spiral groove is etched on the disk and a marble is allowed to travel
in it. The location of a marble along the groove shape is defined in polar
coordinates (which coincide with the center of the disk) as R = R0 e−0.1α ,
where R is the distance to the center of the disk and α is the angle as shown
(thus (R, α) are the polar coordinates of the marble). The disk itself can
rotate around its center with angular speed Ω.
1. We consider first the reference frame of the disk (O, k, l). At the location of the marble on the groove (the black dot), please draw the
basis associated with polar coordinates. Then draw the Frenet basis,
according to what was discussed in lecture and in the book. Are these
two bases the same?
2. Let α be the time derivative of α. Assume α = 0.3 rad/sec, that is, it
˙
˙
5

is constant. Compute the speed of the marble as a function of time.
What happens when time tends towards +∞?
3. In the reference frame of the disk, plot the instantaneous center of
curvature of the trajectory followed by the marble on the groove. Does
the location of the center of curvature change with the speed at which
the marble travels along the groove (that is, for different values of α)?
˙
4. Assume from now on that the disk itself rotates with constant angular
velocity Ω, positive or negative. We would like to plot the location
of the center of curvature of the trajectory on the groove in the nonrotating reference frame (O, i, j) for different values of Ω and α. Assume
˙
that the two bases (i, j) and (k, l) coincide at time t = 0. Assume first
α = 0.3rad/sec and Ω = 0.3rad/sec. In the absolute reference frame
˙
(O, i, j), plot the locus of the trajectory along the groove. Then plot
the location of the instantaneous center of rotation.
5. Repeat the previous question with α = 0.3rad/sec and Ω = −0.3rad/sec.
˙

6
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The Marble moving on a grooved spiral

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