A coin with a center hole is wobbling about a frictionless pole. A torque $$ is

applied to the coin and is always along the central axis hat$()($blue$)$ of the coin. At

the contact point $p$ between the coin and the ground, the coin slides with no

friction in the radial direction, as the blue arrow shows. In the tangent

direction at point $p,$ however, the coin does not slide and maintains a gear$-$like

contact, rolling in the tangent direction. Gravity $g$ exist is the $-$widehat$(K)($red$)$

direction. The coin has mass $m,$ moment of inertia $I_2$ in the hat$()$ direction, and

moment of inertia $I_1$ when measured from the side $($ hat$()$ and hat$(k)$ directions$)$

Derive the dynamic equations of this system using angles $$ and $$ by

answering the following problems:

$(1)$ Describe unit vector widehat$(K)($red$),$ which is fixed in the universe, by means of

coordinate system hat$(),$hat$(),$hat$(k)($blue$),$ which is fixed to the wobbling coin. Note that hat$()$ is

perpendicular to the face of the coin.

$(2)$ Use hat$(),$hat$(),$hat$(k)($blue$)$ to describe vector hat$({)}^{\text{'}}$ and hat$({)}^{\text{'}}($green$),$ which is transformed from

hat$(I),$hat$(J),$widehat$(K)($red$)$ through one rotation of angle $$ about the axis widehat$(K).$

$(3)$ Use hat$(),$hat$(),$hat$(k)($blue$)$ to describe vector hat$(k{)}^{\text{'}\text{'}}($orange$),$ which is transformed from hat$({)}^{\text{'}},$

$j^{\text{'}},$hat$(k{)}^{\text{'}}($green$)$ through one rotation of angle $$ about the axis hat$({)}^{\text{'}}.$

$(4)$ Calculate the velocity of the contact point $p$ at the ground, vec$(v{)}_p,$ which is only in

the $+-$hat$({)}^{\text{'}}($green$)$ direction, using the coordinate system hat$(),$hat$(),$hat$(k)($blue$).$

$(5)$ Calculate the angular velocity of the coin using the coordinate system hat$(),$hat$(),$hat$(k)$

$($blue$).$ Prove that the answer is