Step 1

We are given the acceleration and velocity at the bottom A of the ladder are ${a_A} = 4\;{\rm{ft/}}{{\rm{s}}^2}$ and ${v_A} = 6\;{\rm{ft/s}}$.

We are asked to determine the acceleration of the top B of the ladder and the ladder's angular acceleration.

 
Step 2

The free body diagram of the ladder is shown as:

We have the length of the ladder is $L = 16\;{\rm{m}}$.

We have the angle of the ladder with horizontal is $\theta = 30^\circ $.

We have the vector form of acceleration of A is $\overrightarrow {{a_A}} = - 4i\;{\rm{ft/}}{{\rm{s}}^2}$.

 
Step 3

The formula to calculate the angular velocity of the ladder is,

\[{v_A} = \omega \times L\sin \theta \]
 
Step 4

Substitute the values in the above expression.

\[\begin{array}{c} 6\;{\rm{ft/s}} = \omega \times \left( {16\;{\rm{ft}}\sin 30^\circ } \right)\\ \omega = 0.75\;{\rm{rad/s}} \end{array}\]
 
Step 5

The expression to calculate the position vector of the length the ladder AB is,

\[\overrightarrow {{r_{AB}}} = \left( {L\cos \theta } \right)i + \left( {L\sin \theta } \right)j\]
 
Step 6

Substitute the values in the above expression.

\[\begin{array}{l} \overrightarrow {{r_{AB}}} = \left( {16\cos 30^\circ \;{\rm{ft}}} \right)i + \left( {16\sin 30^\circ \;{\rm{ft}}} \right)j\\ \overrightarrow {{r_{AB}}} = \left( {13.85i + 8j} \right)\;{\rm{ft}} \end{array}\]
 
Step 7

The formula to calculate the acceleration of point B is,

\[\overrightarrow {{a_B}} = \overrightarrow {{a_A}} + \left( {\alpha \times \overrightarrow {{r_{AB}}} } \right) - {\omega ^2}\overrightarrow {{r_{AB}}} \]
 
Step 8

Substitute the values in the above expression.

\[\begin{array}{c} - {a_B}j = \left( { - 4i} \right) + \alpha k \times \left( {13.85i + 8j} \right)\;{\rm{ft}} - {\left( {0.75\;{\rm{rad/s}}} \right)^2}\left( {13.85i + 8j} \right)\;{\rm{ft}}\\ - {a_B}j = \left( { - 4i} \right) + \left| {\begin{array}{*{20}{c}} i&j&k\\ 0&0&\alpha \\ {13.85}&8&0 \end{array}} \right| - 7.8i - 4.5j\\ - {a_B}j = \left\{ {\left( { - 4i} \right) + \left[ \begin{array}{l} i\left( {0 \times 0 - \alpha \times 8} \right) - j\left( {0 \times 0 - \alpha \times 13.85} \right)\\ + k\left( {0 \times 8 - 0 \times 13.85} \right) \end{array} \right] - 7.8i - 4.5j} \right\}\\ - {a_B}j = \left\{ {\left( { - 4i} \right) + \left( { - 8\alpha i + 13.85\alpha j + 0k} \right) - 7.8i - 4.5j} \right\}\\ - {a_B}j = \left( { - 11.80 - 8\alpha } \right)i + \left( {13.85\alpha - 4.5} \right)j \end{array}\]
 
Step 9

On comparing $i$ components in the above expression, we get:

\[\begin{array}{c} \left( { - 11.80 - 8\alpha } \right) = 0\\ \alpha = - 1.475\;{\rm{rad/}}{{\rm{s}}^2} \end{array}\]
 
Step 10

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

\[ - {a_B} = \left( {13.85\alpha - 4.5} \right)\]
 
Step 11

Substitute the value of $\alpha $ in the above expression.

\[\begin{array}{c} - {a_B} = 13.85\left( { - 1.475} \right) - 4.5\\ {a_B} = 24.93\;{\rm{ft/}}{{\rm{s}}^2} \end{array}\]