$6\mathrm{.}21$ Concept: Angular velocity, angular acceleration, and a fixed vector. $($Section $8\mathrm{.}3$ and Hw $6\mathrm{.}20).$

The figure to the right shows a rigid body $B$ in a reference frame $A.$

Orthogonal unit vectors widehat$(b{)}_x,$widehat$(b{)}_y,$widehat$(b{)}_z$ are fixed in $B.$ The following questions

relate to B$\text{'}$s angular velocity in A$.$

Circle expressions for ${?}^A$vec$({)}^B$ for which widehat$(b{)}_z$ remains constant $($fixed$)$ in $A.$

${?}^A$vec$({)}^B=3$widehat$(b{)}_z,{?}^A$vec$({)}^B=(3+t)widehat(b{)}_z,{?}^A$vec$({)}^B=3$widehat$(b{)}_x+4$widehat$(b{)}_y,{?}^A$vec$({)}^B=4$widehat$(b{)}_y+t{b}_z$

Circle the expressions for ${?}^A$vec$({)}^B$ that remain constant $($fixed$)$ in $A.$

${?}^A$vec$({)}^B=3$widehat$(b{)}_z,{?}^A$vec$({)}^B=(3+t)widehat(b{)}_z,{?}^A$vec$({)}^B=3$widehat$(b{)}_x+4$widehat$(b{)}_y,{?}^A$vec$({)}^B=4$widehat$(b{)}_y+$twidehat$(b{)}_z$

Complete the calculations below that help verify your answers to the previous two questions.

$\frac{{?}^A\text{d}\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}(b{)}_z}{dt}={?}_{(8\mathrm{.}1)},+,\frac{{?}^A\text{d}\text{v}\text{e}\text{c}(A{)}^A}{dt}={?}_{(8\mathrm{.}1)},+$

Circle those expressions that result in ${?}^A$vec$({)}^B($B$\text{'}$s angular acceleration in A$)$ being non$-$zero.

${?}^A$vec$({)}^B=3$widehat$(b{)}_z,{?}^A$vec$({)}^B=(3+t)widehat(b{)}_z,{?}^A$vec$({)}^B=3$widehat$(b{)}_x+4$widehat$(b{)}_y,{?}^A$vec$({)}^B=4$widehat$(b{)}_y+$twidehat$(b{)}_z$