Step 1

We are given a ball $B$ which is rolling along the slot in the disk.

The angular acceleration of the disk is, $\alpha = 3{\rm{ rad/}}{{\rm{s}}^2}$.

The angular velocity of the disk is, $\omega = 6{\rm{ rad/s}}$.

The radius of the disk is, $r = 0.8{\rm{ m}}$.

The velocity of the ball rolling in the slot is, ${v_{BO}} = 600{\rm{ mm/s}}$.

The acceleration of the ball rolling in the slot is, ${a_{BO}} = 150{\rm{ mm/}}{{\rm{s}}^2}$.

We are asked to determine the velocity and acceleration of the ball.

 
Step 2

The velocity diagram of the link $AB$ is given below:

At the instant, the distance of the ball from the origin $O$ is given by,

\[{r_{BO}} = \left( {0.4{\bf{i}}} \right){\rm{ m}}\]

The velocity of the ball $B$ rolling along the slot with respect to the origin $O$ is given by,

\[{v_{BO}} = \left( {0.6{\bf{i}}} \right){\rm{ m/s}}\]

At the axis passing through origin $O$, the velocity of the disc at $O$ is given by,

\[{v_O} = 0{\rm{ m/s}}\]

And, the acceleration of the disc at $O$ is given by,

\[{a_O} = 0{\rm{ m/}}{{\rm{s}}^2}\]
 
Step 3

The disk is rotating about the $z$ axis, so the angular velocity of the disk about $z$ axis is given by,

\[{\omega _z} = \left( {{\rm{6}}{\bf{k}}} \right){\rm{ rad/s}}\]

Similarly, the angular acceleration of the disk about $z$ axis is given by,

\[{\alpha _z} = \left( {{\rm{3}}{\bf{k}}} \right){\rm{ rad/}}{{\rm{s}}^2}\]

The expression for the vector form of the net velocity of the ball $B$ is given by,

\[\begin{array}{c} {{\rm{v}}_B} = {v_O} + \omega {{\rm{r}}_{BO}} + {v_{BO}}\\ {{\rm{v}}_B} = 0 + \left( {6{\bf{k}}} \right) \times \left( {0.4{\bf{i}}} \right) + \left( {0.6{\bf{i}}} \right)\\ {{\rm{v}}_B} = \left( {0.6{\bf{i}} + 2.4{\bf{j}}} \right){\rm{ m/s}} \end{array}\]
 
Step 4

Similarly, the vector form for the acceleration of the ball $B$ rolling along the slot is given by,

\[\begin{array}{l} {{\rm{a}}_B} = {{\rm{a}}_O} + {\alpha _z}{r_{BO}} + {\omega _z}^2{r_{BO}} + 2{\omega _z}{v_{BO}} + {{\rm{a}}_{BO}}\\ {{\rm{a}}_B} = 0 + \left( {{\rm{3}}{\bf{k}}} \right) \times \left( {0.4{\bf{i}}} \right){\rm{ + }}\left( {{\rm{6}}{\bf{k}}} \right) \times \left[ {\left( {{\rm{6}}{\bf{k}}} \right) \times \left( {0.4{\bf{i}}} \right)} \right] + 2\left( {{\rm{6}}{\bf{k}}} \right) \times \left( {{\rm{0}}{\rm{.6}}{\bf{i}}} \right) + \left( {0.15{\bf{i}}} \right)\\ {{\rm{a}}_B} = 0 + 1.2{\bf{j}} - 14.4{\bf{i}} + 7.2{\bf{j}} + 0.15{\bf{i}}\\ {{\rm{a}}_B} = \left( { - 14.25{\bf{i}} + 8.40{\bf{j}}} \right){\rm{ m/}}{{\rm{s}}^2} \end{array}\]