A particle of mass m rotates in a circle of radius a with a...

A particle of mass $m$ rotates in a circle of radius $a$ with a uniform angular speed $\omega_{0}$. It is viewed from a frame rotating about the $z$ -axis with a uniform angular speed $\omega$. The centrifugal force on the particles is: (a) $m \omega^{2} a$ (b) $m \omega_{0}^{2} a$ (c) $m\left(\frac{\omega+\omega_{0}}{2}\right)^{2} a$ (d) $m \omega \omega_{0}$

A particle of mass $m$ rotates in a circle of radius $a$ with a uniform angular speed $\omega_{0}$. It is viewed from a frame rotating about the $z$ -axis with a uniform angular speed
$\omega$. The centrifugal force on the particles is:
(a) $m \omega^{2} a$
(b) $m \omega_{0}^{2} a$
(c) $m\left(\frac{\omega+\omega_{0}}{2}\right)^{2} a$
(d) $m \omega \omega_{0}$

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Key Concepts

Uniform Circular Motion

Uniform circular motion refers to motion in a circle at constant speed. In an inertial frame, the required centripetal force keeping the particle on a circular path is m(??)^2a, where ?? is the angular speed. However, when observed from a rotating frame, the apparent forces (including the centrifugal force) are determined by the frame’s own rotation, illustrating the difference between inertial forces and fictitious forces.

Rotating Reference Frame

A rotating reference frame is a non?inertial frame whose axes rotate with respect to an inertial frame. Observers in such frames must account for extra apparent forces (fictitious forces) when applying Newton’s laws, because the rotating frame accelerates relative to an inertial frame. This concept helps explain why objects appear to experience forces like the centrifugal force even when no physical interaction is present.

Fictitious (Inertial) Forces – Centrifugal Force

In a rotating frame, fictitious forces must be introduced to explain the observed motion. The centrifugal force is one such force, mathematically expressed as m(?_frame)^2r, where ?_frame is the rotational speed of the frame and r is the distance from the axis. It acts radially outward and entirely depends on the rotation rate of the frame rather than the particle’s motion in an inertial frame.

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