Question

A particle of mass \( m \) fixed in space is observed from a frame rotating about its \( z \)-axis with angular speed \( \omega \). The particle is in the frame’s \( xy \)-plane at a distance \( R \) from its origin. If the Coriolis and centrifugal forces on the particle are \( F_{\text{COR}} \) and \( F_{\text{CF}} \), respectively, then (all the symbols have their standard meaning and refer to the rotating frame),

• A. \( F_{\text{COR}} + F_{\text{CF}} = 0 \)
• B. \( F_{\text{COR}} = -m \omega^2 R \hat{r} \)
• C. \( F_{\text{COR}} = -2 m \omega R \hat{\theta} \)
• D. \( F_{\text{CF}} = -m \omega^2 R \hat{r} \)

Hint:

The Coriolis force is perpendicular to the velocity of the particle and the centrifugal force is directed radially outward in a rotating frame.

Answer 1

The Correct Option is C

Step 1: Understanding Coriolis and centrifugal forces.
The Coriolis force is given by \( F_{\text{COR}} = -2 m \omega R \hat{\theta} \), which acts perpendicular to the velocity of the particle. The centrifugal force acts radially outward and is given by \( F_{\text{CF}} = m \omega^2 R \hat{r} \). These forces result from the rotating reference frame.
Step 2: Conclusion.
Thus, the correct answer is option (C).