The diagram at right shows a solid disk that is mounted so that it can rotate about its center. Four identical thin rods are welded to the disk. The four rods form a perfect square and the square is centered on the disk. The diagonal of the square is equal to the diameter of the disk. A string is wrapped around the disk and a rock hangs from its end. Use the following notation:

Mass of disk: $4$ M

Mass of each rod: $\backslash (3$ M $\backslash )$

Mass of rock: $\backslash ($ M $\backslash )$

Length of each rod: L

$1.$ What is the radius of the disk in terms of L $?$

Radius of Disk $=$

$2.$ Distance between the center of the disk and the center of each rod?

Distance $=$

$3.$ Moment of inertia of the disk$/$four rod system?

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I$\_\{\backslash$text $\{$disk$/$four rod system $\}\}=$

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$4.$ Draw an extended freebody diagram of the disk$/$four rod system on a piece of paper.

$5.$ Draw a regular freebody diagram for the rock on a piece of paper.

The diagrams you drew should contain the following forces:

$\backslash (\backslash$mathrm$\{$T$\}=\backslash )$ tension in the string $\backslash (\backslash$quad T$\_\{\backslash$text $\{$pivot $\}\}=\backslash )$ tension supporting the disks's pivot

$\backslash ($ W$\_\{\backslash$text $\{$pulley $\}\}=\backslash )$ weight of disk and four rods $\backslash ($ W$\_\{\backslash$text $\{$rock $\}\}=\backslash )$ weight of the rock

$6.$ Write the rotational Newton's Second Law for the disk$/$four rod system. Your equation should be expressed in terms of the forces above as well as $\backslash ($ M$,$ L $\backslash )$ and $\backslash (\backslash$alpha $\backslash ),$ the magnitude of the angular acceleration of the disk$/$four rod system.

$7.$ Write Newton's Second Law for the rock.

$8.$ Write an expression for the angular acceleration of the disk$/$four rod system, $\backslash (\backslash$alpha $\backslash ),$ in terms of the acceleration of the rock, $\backslash ($ a $\backslash )$ and $\backslash ($ L $\backslash ).$

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$\backslash$alpha$=$

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$9.$ Solve the system of equations for the acceleration of the rock. Express your answer in terms of M$,$ L and$/$or g$.$

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a$=$

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$10.$ Derive an expression for the amount of time it takes the disk$/$four rod system to rotate $3$ complete revolutions if it starts from rest. Express your answer in terms of $\backslash ($ M$,$ L $\backslash )$ and$/$or g $.$

$\backslash [$

$\backslash$Delta t$=$

$\backslash ]$