FrontierScience / bc8a05ac-9db5-47b4-8618-84aa4a46466a / Consider a particle of mass `m`, that is constrained to move on a spherical shell with radius `R`. The spherical shell is rotating with angular velocity `ω`. Initially this particl…

Consider a particle of mass `m`, that is constrained to move on a spherical shell with radius `R`. The spherical shell is rotating with angular velocity `ω`.  Initially this particle is at the equator and is initially projected northward perpendicular to the equator (in the rotating reference frame of the sphere) at a surface relative velocity of `v`. Ignore any resistive forces. Denote the greatest latitude of the particle as `θ_(max)`. Assume `θ_(max)≪1` so that the small angle approximations `sin(θ)=θ` and `cos(θ)=1` can be used. Denote the latitude of the particle at time `t` as `θ(t)`. Find the time period `T` of `θ(t)`.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.

Consider a particle of mass `\( m \)`, that is constrained to move on a spherical shell with radius `\( R \)`. The spherical shell is rotating with angular velocity `\( \omega \)`.  Initially this particle is at the equator and is initially projected northward perpendicular to the equator (in the rotating reference frame of the sphere) at a surface relative velocity of `\( v \)`. Ignore any resistive forces. Denote the greatest latitude of the particle as `\( \theta_{max} \)`. Assume `\( \theta_{max} \ll 1 \)` so that the small angle approximations `\( \sin(\theta) = \theta \)` and `\( \cos(\theta) = 1 \)` can be used. Denote the latitude of the particle at time `\( t \)` as `\( \theta(t) \)`. Find the time period `\( T \)` of `\( \theta(t) \)`.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.