Step 1

We are given the angular velocity of disk about the z-axis is ${\omega _z} = 0.5\;{\rm{rad/s}}$.

We are asked to determine the velocity and the acceleration of point A on the disk.

 
Step 2

The diagram of the system is shown as:

We have the radius of disk is $r = 150\;{\rm{mm}}$.

We have the distance of disk end from x-axis is $d = 300\;{\rm{mm}}$.

 
Step 3

The expression to calculate the angular velocity of the disk is,

\[\omega = {\omega _s} + {\omega _z}\]

Here, ${\omega _s}$ is the spinning angular velocity.

 
Step 4

Substitute the values in the above expression.

\[\begin{array}{c} - \omega j = - {\omega _s}\cos 30^\circ j - {\omega _s}\sin 30^\circ k + 0.5k\\ - \omega j = - 0.866{\omega _s}j + \left( { - 0.5{\omega _s} + 0.5} \right)k \end{array}\]
 
Step 5

On equating $k$ components in the above expression, we get:

\[\begin{array}{c} \left( { - 0.5{\omega _s} + 0.5} \right) = 0\\ {\omega _s} = 1\;{\rm{rad/s}} \end{array}\]
 
Step 6

On equating $j$ components in the above expression, we get:

\[\begin{array}{c} - \omega = - 0.866{\omega _s}\\ - \omega = - 0.866\left( {1\;{\rm{rad/s}}} \right)\\ \omega = 0.866\;{\rm{rad/s}} \end{array}\]
 
Step 7

The formula to calculate the angular acceleration of the disk is,

\[\alpha = {(\dot \omega )_{xyz}} + {\omega _z} \times \omega \]

Here, ${(\dot \omega )_{xyz}}$ is the rate of angular velocity along fixed frame XYZ and its value is zero.

 
Step 8

Substitute the values in the above expression.

\[\begin{array}{c} \alpha = 0 + 0.5k \times - 0.866j\\ \alpha = 0.433i\;{\rm{rad/}}{{\rm{s}}^{\rm{2}}} \end{array}\]
 
Step 9

The formula to calculate the position vector of A is,

\[\begin{array}{c} {r_A} = \left[ {\left( {0.3 - 0.3\cos 60^\circ } \right)j + 0.3\sin 60^\circ k} \right]\\ {r_A} = \left( {0.15j + 0.2598k} \right)\;{\rm{m}} \end{array}\]
 
Step 10

The formula to calculate the velocity at A on the disk is,

\[{v_A} = \omega \times {r_A}\]
 
Step 11

Substitute the values in the above expression.

\[\begin{align} & {{v}_{A}}=-0.866j\times (0.15j+0.2598k) \\ & {{v}_{A}}=-0.225i\,\text{m/s} \\ \end{align}\]
 
Step 12

The formula to calculate the acceleration at A on the disk is,

\[a = \alpha \times {r_A} + \omega \left( {\omega \times {r_A}} \right)\]
 
Step 13

Substitute the values in the above expression.

\[\begin{array}{c} a = 0.433i \times \left( {0.15j + 0.2598k} \right) + 0.866\left[ {0.866 \times \left( {0.15j + 0.2598k} \right)} \right]\\ a = 0.06495k - 0.1125j + 0.866j \times \left( {0.2249i} \right)\\ a = 0.06495k - 0.1125j - 0.1947k\\ a = ( - 0.1125j - 0.13k){\rm{m/}}{{\rm{s}}^2} \end{array}\]