A mass m is free to move without friction in a hoop of radius R, which has no mass. The hoop rotates with a constant angular velocity ω around its vertical diameter (see the figure below). (a) Write the Lagrangian of the system as a function of θ. (b) Find the equation of motion for mass m. (c) Determine the equilibrium angles, that is, the conditions for which ˙θ = ¨θ = 0 . Consider two cases: ω² ≥ g/R and ω² < g/R.

A mass m is free to move without friction in a hoop of radius R, which has no mass. The hoop rotates with a constant angular velocity ω around its vertical diameter (see the figure below). (a) Write the Lagrangian of the system as a function of θ. (b) Find the equation of motion for mass m. (c) Determine the equilibrium angles, that is, the conditions for which ˙θ = ¨θ = 0 . Consider two cases: ω² ≥ g/R and ω² < g/R.

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Question

A mass m is free to move without friction in a hoop of radius R, which has no mass. The hoop rotates with a constant angular velocity ω around its vertical diameter (see the figure below).

(a) Write the Lagrangian of the system as a function of θ.

(b) Find the equation of motion for mass m.

Consider two cases: ω² ≥ g/R and ω² < g/R.

Definition Definition Rate of change of angular displacement. Angular velocity indicates how fast an object is rotating. It is a vector quantity and has both magnitude and direction. The magnitude of angular velocity is represented by the length of the vector and the direction of angular velocity is represented by the right-hand thumb rule. It is generally represented by ω.

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