Figure $1$ depicts a half$-$disk with mass m and mass moment of inertia J$\_$g$.$ G is the center of mass of the half disk. The disk rotates without slipping. Point O is a fixed point and can be used as the center of a fixed reference frame. Point C is the contact point of the disk with the surface. Obtain the motion equation of the half$-$disk. $($Hint: start by writing the coordinates of the point G and then find the velocity and acceleration of the center of mass$)$

Problem $1.(20$ points$)$

Figure $1$ depicts a half$-$disk with mass m and mass moment of inertia J$\_($G$).$G is the center of

mass of the half disk. The disk rotates without slipping. Point O is a fixed point and can be

used as the center of a fixed reference frame. Point C is the contact point of the disk with the

surface. Obtain the motion equation of the half$-$disk. $($Hint: start by writing the coordinates of

the point G and then find the velocity and acceleration of the center of mass$)$