Problem 1. Circular motion in the rotating frame. Check consistency. (8 points)

Question

Problem 1. Circular motion in the rotating frame. Check consistency. (8 points) Consider a mass point M which is in uniform circular motion with radius R in the horizontal plane. The angular velocity is omega = omega z in the inertial Lab frame. What is the magnitude of the centripetal force fc acting on M in the Lab frame? Now let us describe the same motion in a frame rotating with the angular velocity ohm = ohm z with respect to the Lab frame. The rotation center coincides with the circular motion of M. What is the angular velocity of the uniform circular motion of M in the rotating frame? What are the magnitudes and directions of the two inertial forces for the motion of M in this rotating frame? Combining f and inertial forces together, show that they together are just the centripetal force for the motion you have described in 2).

Explanation / Answer

a) Angular velocity and angular acceleration remain constant (fixed at zero when there is constant angular velocity). That's the meaning of "uniform".

b) The tangential velocity vector is continuously changing direction and, consequently, there is a linear acceleration due to the directional changes. The direction of the linear acceleration is always centripetal, inward on the radius of rotation. Which means, although its magnitude is not changing, its direction is as the radius rotates around the axis of turn.

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