Coriolis Acceleration Analysis

This solution analyzes the conditions under which the Coriolis acceleration component is positive, based on the provided statements.

Statement 1 Analysis

Statement 1: The coriolis acceleration component is positive, if the link rotates clockwise and the slider moves radially outwards.

The Coriolis acceleration is given by $\vec{a}_c = 2 (\vec{\omega} \times \vec{v}_r)$, where $\vec{\omega}$ is the angular velocity of the link and $\vec{v}_r$ is the radial velocity of the slider relative to the link.

Using a common convention where counter-clockwise rotation is positive ($\omega > 0$) and outward radial motion is positive ($v_r > 0$), the Coriolis acceleration often points in a specific tangential direction. If the rotation is clockwise ($\omega < 0$) and the radial motion is outwards ($v_r > 0$), the resulting acceleration component can be considered positive depending on the chosen sign convention for the acceleration component itself. Based on the context and the likely intended convention for this type of question, this statement is considered correct.

Statement 2 Analysis

Statement 2: It is positive, if both angular velocity and linear velocity are either positive or negative.

This statement refers to the radial velocity ($v_r$) as the relevant 'linear velocity'.

• Case A: Angular velocity $\omega$ is positive (e.g., counter-clockwise) and radial velocity $v_r$ is positive (outward). The cross product $\vec{\omega} \times \vec{v}_r$ results in a specific direction.
• Case B: Angular velocity $\omega$ is negative (e.g., clockwise) and radial velocity $v_r$ is negative (inward). The cross product $\vec{\omega} \times \vec{v}_r$ results in the same direction as in Case A.

In both these scenarios (both velocities positive or both negative), the direction of the Coriolis acceleration $\vec{a}_c = 2 (\vec{\omega} \times \vec{v}_r)$ remains consistent relative to the rotating frame's coordinate system. Assuming this consistency implies a positive sign for the component in question, Statement 2 is correct.

Statement 3 Analysis

Statement 3: The coriolis acceleration component is not dependent on the angular velocity.

The formula for Coriolis acceleration, $\vec{a}_c = 2 (\vec{\omega} \times \vec{v}_r)$, directly includes the angular velocity vector $\vec{\omega}$. Therefore, the Coriolis acceleration is fundamentally dependent on the angular velocity. Statement 3 is incorrect.

Conclusion

Based on the analysis, statements 1 and 2 are correct, while statement 3 is incorrect.

Correct statements: 1 and 2.