Section $1\mathrm{.}7$

$1\mathrm{.}12$ The absolute position, velocity, and acceleration of $O$ are

$r_0=-22$hat$()+22$hat$()+22$hat$(k)(m)$

$v_0=7$hat$(I)+9$hat$(J)+4$hat$(K)(\frac{m}{s})$

$a_0=3$hat$(I)-7$hat$(J)+4$hat$(K)(\frac{m}{s^2})$

The angular velocity and acceleration of the moving frame are

$=-0\mathrm{.}8$hat$(I)+0\mathrm{.}7$hat$(J)+0\mathrm{.}4$hat$(K)(ra\frac{d}{s}),{}^{}=-0\mathrm{.}4$hat$(I)+0\mathrm{.}9$hat$(J)-1\mathrm{.}0$hat$(K)(ra\frac{d}{s^2})$

The unit vectors of the moving frame are

hat$(i)=-0\mathrm{.}15670$hat$(I)-0\mathrm{.}31235$hat$(J)+0\mathrm{.}93704$hat$(K)$

hat$(j)=-0\mathrm{.}12940$hat$(I)+0\mathrm{.}94698$hat$(J)+0\mathrm{.}29409$hat$(K)$

hat$(k)=-0\mathrm{.}97922$hat$(I)-0\mathrm{.}075324$hat$(J)-0\mathrm{.}18831$hat$(K)$

The absolute position of $P$ is

$r=51$hat$(I)-45$hat$(J)+36$hat$(K)(m)$

The velocity and acceleration of $P$ relative to the moving frame are

$v_{\text{r}\text{e}\text{l}}=31$hat$(i)-68$hat$(j)-77$hat$(k)(\frac{m}{s}),a_{\text{r}\text{e}\text{l}}=2$hat$(i)-6$hat$(j)+5$hat$(k)(\frac{m}{s^2})$

Calculate the absolute velocity $v_P$ and acceleration $a_P$ of $P.$

$\{$Ans$.$: $v_P=156\mathrm{.}4a_v(\frac{m}{s}),a_v=0\mathrm{.}7790$hat$(I)-0\mathrm{.}3252$hat$(J)+0\mathrm{.}5360$hat$(K)$

$a_P=85\mathrm{.}13a_a(\frac{m}{s^2}),\{$:hat$(u{)}_a=-0\mathrm{.}3229(hat(I))+0\mathrm{.}8284(hat(J))-0\mathrm{.}4576(hat(K))\}$