Crank AB 5 rad/s, piston velocity

Assume the crank $AB$ turns counter-clockwise at $\omega_{AB}=5$ rad/s, with $A$ as origin, $B$ above and to the right of $A$, and the slider line $0.15$ m above $A$ with $C$ to the right of $B$ (the magnitudes of the answers are the same for the other sense of rotation).

Geometry

$$\vec r_{B/A} = 0.6(\cos30^\circ\hat i + \sin30^\circ\hat j) = (0.5196\,\hat i + 0.3000\,\hat j)\ \text{m}$$

$B$ is $0.300 - 0.150 = 0.150$ m above the slider line, so $\sin\phi = 0.150/0.300 = 0.5$ and $\phi=30^\circ$ (link $BC$ slopes down to the right):

$$\vec r_{C/B} = 0.3(\cos30^\circ\hat i - \sin30^\circ\hat j) = (0.2598\,\hat i - 0.1500\,\hat j)\ \text{m}$$

Velocities

$$\vec v_B = \omega_{AB}\hat k\times\vec r_{B/A} = 5\hat k\times(0.5196\hat i + 0.3000\hat j) = (-1.500\,\hat i + 2.598\,\hat j)\ \text{m/s}$$ $$\vec v_C = \vec v_B + \omega_{BC}\hat k\times\vec r_{C/B} = \vec v_B + \omega_{BC}(0.1500\,\hat i + 0.2598\,\hat j)$$

The slider moves horizontally, so the $\hat j$ component of $\vec v_C$ is zero:

$$0 = 2.598 + 0.2598\,\omega_{BC} \;\Rightarrow\; \omega_{BC} = -10.00\ \text{rad/s}\ \text{(clockwise)}$$ $$v_C = -1.500 + 0.1500(-10.00) = -3.00\ \text{m/s}$$

Answer: $v_C = 3.00$ m/s (horizontal, towards $A$), $\omega_{BC} = 10.00$ rad/s clockwise.

Coriolis acceleration

Coriolis acceleration is the extra acceleration a point has when it moves relative to a rotating (moving) frame or body. If a point $P$ moves with relative velocity $\vec v_{rel}$ on a body rotating with angular velocity $\vec\omega$, then

$$\vec a_P = \vec a_{P'} + \vec a_{rel} + 2\vec\omega\times\vec v_{rel}$$

where $\vec a_{P'}$ is the acceleration of the coincident point of the rotating body and $\vec a_{rel}$ is the acceleration of $P$ relative to the body. The term $\vec a_c = 2\vec\omega\times\vec v_{rel}$ is the Coriolis acceleration. Its magnitude is $2\omega v_{rel}$ and its direction is that of $\vec v_{rel}$ rotated through $90^\circ$ in the sense of $\omega$. It exists only when the point slides on the rotating body (for example a collar sliding on a rotating arm, or a slider in a slotted rotating link); it is zero for two points fixed in the same rigid body, such as the crank pin and piston pin of a connecting rod.