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