Step 1

We are given the angular velocity of shaft in z-axis is ${\omega _z} = 4\;{\rm{rad/s}}$, and the angular acceleration of the shaft along z-axis is ${\left( {{{\dot \omega }_1}} \right)_{xyz}} = 2\;{\rm{rad/}}{{\rm{s}}^2}$.

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

 
Step 2

The diagram of the system is shown as:

We have the radius of the disk is $r = 0.1\;{\rm{m}}$.

We have the distance between disk and shaft end is $l = 0.5\;{\rm{m}}$.

We have the position vector of point A on the disk is ${r_A} = \left( {0.5i + 0.1k} \right)\;{\rm{m}}$.

 
Step 3

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

\[\omega = {\omega _1} + {\omega _2}\]
 
Step 4

Substitute the values in the above expression.

\[\omega = \frac{5}{{\sqrt {26} }}\omega i - \frac{1}{{\sqrt {26} }}\omega k\] … (1)

Now, substitute the value of $\omega $ component on the left side, we get,

\[( - 4k + {\omega _2}i) = \frac{5}{{\sqrt {26} }}\omega i - \frac{1}{{\sqrt {26} }}\omega k\]
 
Step 5

On comparing $k$ coefficients in the above expression.

\[\begin{array}{c} - 4 = - \frac{1}{{\sqrt {26} }}\omega \\ \omega = 4\sqrt {26} \;{\rm{rad/s}} \end{array}\]
 
Step 6

On comparing $i$ coefficients in the above expression.

\[\begin{array}{c} {\omega _2} = \frac{5}{{\sqrt {26} }}\omega \\ {\omega _2} = \frac{5}{{\sqrt {26} }}\left( {4\sqrt {26} } \right)\\ {\omega _2} = 20\;{\rm{rad/s}} \end{array}\]
 
Step 7

Substitute the value of $\omega $ on the right side in the equation (1).

\[\begin{array}{c} \omega = \frac{5}{{\sqrt {26} }}\left( {4\sqrt {26} } \right)i - \frac{1}{{\sqrt {26} }}\left( {4\sqrt {26} } \right)k\\ \omega = \left( {20i - 4k} \right)\;{\rm{rad/s}} \end{array}\]
 
Step 8

The formula to calculate the rotating angular acceleration with reference to xyz rotating frame is,

\[\frac{{{{\left( {{{\dot \omega }_2}} \right)}_{xyz}}}}{{{{\left( {{{\dot \omega }_2}} \right)}_{xyz}}}} = \frac{l}{r}\]
 
Step 9

Substitute the values in the above expression.

\[\begin{array}{c} \frac{{{{\left( {{{\dot \omega }_2}} \right)}_{xyz}}}}{{{{\left( {{{\dot \omega }_2}} \right)}_{xyz}}}} = \frac{{0.5}}{{0.1}}\\ {\left( {{{\dot \omega }_2}} \right)_{xyz}} = 5{\left( {{{\dot \omega }_1}} \right)_{xyz}} \end{array}\]
 
Step 10

The formula to calculate the rotating angular velocity with reference to xyz rotating frame is,

\[\Omega = {\omega _1}\]
 
Step 11

Substitute the values in the above expression.

\[\Omega = - 4k\;{\rm{rad/s}}\]
 
Step 12

The formula to calculate the rotating angular acceleration of the frame is,

\[{\dot \omega _2} = {\left( {{{\dot \omega }_2}} \right)_{xyz}} + \Omega \times {\omega _2}\]
 
Step 13

Substitute the values in the above expression.

\[\begin{array}{c} {{\dot \omega }_2} = 10i + \left( { - 4k} \right) \times \left( {20i} \right)\\ {{\dot \omega }_2} = \left( {10i - 80j} \right)\;{\rm{rad/}}{{\rm{s}}^2} \end{array}\]
 
Step 14

The formula to calculate the rotating angular acceleration of the frame is,

\[{\dot \omega _1} = {\left( {{{\dot \omega }_1}} \right)_{xyz}} + {\omega _1} \times {\omega _1}\]
 
Step 15

Substitute the values in the above expression.

\[\begin{array}{l} {{\dot \omega }_1} = - 2k + \left( {\frac{5}{{\sqrt {26} }}\omega i \times \frac{5}{{\sqrt {26} }}\omega i} \right)\\ {{\dot \omega }_1} = \left( { - 2k} \right)\;{\rm{rad/}}{{\rm{s}}^2} \end{array}\]
 
Step 16

The formula to calculate the final rotating angular acceleration of the shaft is,

\[\alpha = {\dot \omega _1} + {\dot \omega _2}\]
 
Step 17

Substitute the values in the above expression.

\[\begin{array}{c} \alpha = \left( { - 2k} \right)\;{\rm{rad/}}{{\rm{s}}^2} + \left( {10i - 80j} \right)\;{\rm{rad/}}{{\rm{s}}^2}\\ \alpha = \left( {10i - 80j - 2k} \right)\;{\rm{rad/}}{{\rm{s}}^2} \end{array}\]
 
Step 18

The formula to calculate the velocity of point A on the disk is,

\[v = \omega \times {r_A}\]
 
Step 19

Substitute the values in the above expression.

\[\begin{array}{c} v = \left( {20i - 4k} \right)\;{\rm{rad/s}} \times \left( {0.5i + 0.1k} \right)\;{\rm{m}}\\ v = \left( { - 4j} \right)\;{\rm{m/s}} \end{array}\]
 
Step 20

The formula to calculate the acceleration of point A on the disk is,

\[{a_A} = \left( {\alpha \times {r_A}} \right) + \left( {\omega \times v} \right)\]
 
Step 21

Substitute the values in the above expression.

\[\begin{array}{c} {a_A} = \left[ {\left( {10i - 80j - 2k} \right) \times \left( {0.5i + 0.1k} \right)} \right] + \left[ {\left( {20i - 4k} \right) \times \left( { - 4j} \right)} \right]\\ {a_A} = \left( { - 24i - 2j - 40k} \right)\;{\rm{m/}}{{\rm{s}}^2} \end{array}\]