Problem $6$

Arm BCD rotates about the $z$ axis with angular velocity ${}_1$ of $5ra\frac{d}{s}$ which increases at the rate of $10ra\frac{d}{s^2}.$ The disk also rotates about BC with a constant angular velocity ${}_2$ of $4ra\frac{d}{s}.$ System $"1"$ is the fixed system $xYZ$ as shown. System $"$a$"$ is created as vec$({)}_1$ rotates about the $Z$ axis with angle $.$ System $"$b$"$ is created as vec$({)}_2$ rotates about the $y$ axis with respect to System $"$a$"$ with angle $.$ Evaluate the following quantities when $8=15$ and $=20.$

a$.$ Define each angular velocity $($ vec$({)}_1$ and vec$({)}_2)$ in terms of its local system. $($Notice that each angular velocity is local to TwO systems. Partial Answer: vec$({)}_1=5$hat$(K{)}_{\text{I}}\frac{\text{r}\text{a}\text{d}}{s}$ and vec$({)}_1=5$hat$(k{)}_{a}ra\frac{d}{s})$

b$.$ Use transformation matrices to transform vec$({)}_1$ and vec$({)}_2$ from their local systems to the global System "I" $($XYZ$).$ Use these to find the total angular velocity of the rotating disk in System "I". $($Answer: vec$({)}_{\text{I}}=-1\mathrm{.}37$hat$(l)+3\mathrm{.}76$hat$()+5$hat$(R)ra\frac{d}{s})$

c$.$ Notice that the disk is in "fixed point rotation" about point C$.$ Define a vector vec$(r{)}_{\frac{A}{C}}$ in its local system $($System $"b^{\text{'}}{x}_b{b}_b{z}_b,$ which rotates with angular velocity vec$({)}_2=$vec$({)}^{}).$ Find the velocity of point A in terms of System $"$b$".($Answer: vec$(V{)}_{A_b}=-\mathrm{.}58$hat$()+\mathrm{.}72$hat$(j)-\mathrm{.}755$hat$(k)\frac{m}{s})$

d$.$ Use transformation matrices to find the velocity of A in the inertial system $($System "I" XYZ$).($Answer: vec$(V{)}_{A_{\text{I}}}=-\mathrm{.}96$hat$()+\mathrm{.}42$hat$()-\mathrm{.}58$hat$(k)\frac{m}{s})$

e$.$ Show how to transform vec$(V{)}_{A_l}$ back to the $"$b$"$ system. Verify that your answer matches what you got in part c $.$

f$.$ Draw a sketch of the approximate position of the device after it has moved to position $=15*$ and $=20.$ Include the local axes and point A $.$