Answer
$\omega=-4rad/s$
$v_P=4.838m/s \leftarrow$
Work Step by Step
We can determine the required velocity and angular velocity as follows:
We know that
$v_D=v_B+\omega\times r_{D/B}$
$\implies -v_Dcos30\hat i-v_Dsin30\hat j=(-2.4cos30\hat i+2.4sin 30\hat j)+(\omega\hat k)(0.6\hat i)$
$\implies -v_Dcos30\hat i-v_Dsin30\hat j=-2.4cos 30\hat i +(2.4sin30+0.6\omega)\hat j$
Comparing $i$ components on both sides, we obtain:
$-v_Dcos30=-2.4cos30$
$\implies v_D=2.4m/s$
and for $j$componetnts
$-v_Dsin30=2.4sin30+0.6\omega$
This simplifies to:
$\omega=-4rad/s$
The relative velocity between $B$ and $P$ is given as
$v_P=v_B+\omega\times r_{P/B}$
We plug in the known values to obtain:
$v_P=-2.4cos30\hat i+2.4sin30\hat j+(-4\hat k)\times (0.3\hat i-2\times \frac{0.3}{cos30}\hat j)$
$\implies v_{Px}\hat i+v_{Py}\hat j=-2.4cos30\hat i+2.4sin30\hat j+(-4\hat k)\times (0.3\hat i-0.69\hat j)=(-2.4cos30-2.76)\hat i+(2.4sin30-1.2)\hat j$
Comparing $i$ components on both sides, we obtain:
$v_{Px}=-4.838m/s$
and for $j$ components
$v_{Py}=0$
$\implies v_P=4.838m/s \leftarrow$
