FrontierScience / afb2ebc7-421e-4e61-8073-7c4393aa9855 / Consider an object consisting of a particle with mass `M` connected to two particles, each with mass `m`, via two variable-length, massless arms, each of length `s`. This object is…

Consider an object consisting of a particle with mass `M` connected to two particles, each with mass `m`, via two variable-length, massless arms, each of length `s`. This object is restricted to move on the surface of a sphere with a radius `R`, where `s≪R`. A local Cartesian coordinate system `XY` is established on the sphere. At time `t≤0`, the object is at position `(X,Y)=(0,0)` with both arms in a retracted state (`s=0`) and is at rest. When `t>0`, the object begins to move, with both arms extending to the same length and symmetrically opposite in direction to the X-axis. Let the angle between the two arms be `2θ`, then the relative position of one of the particles with mass `m` to the particle of mass `M` is `(x,y)=s×(cosθ,sinθ), and the other s×(cosθ,−sinθ)`. As the object moves, `(x,y)` undergoes periodic changes as described:

`\((x,y)\colon(0,0)\to\left(0,L_{y}\right)\to\left(L_{x},L_{y}\right)\to\left(L_{x},0\right)\to(0,0)\)`

If \\((x,y)\\) to the next of the 4 states with frequency \\(f\\), find the (signed) average velocity `⟨V⟩` of the object's motion, where positive \\(V\\) is in the direction of the positive \\(X\\) axis. Assuming `m≪M` and `L_(x),L_(y)≪R`, in the calculations, you only need to take the non-zero lowest order terms of `m/M`, `L_(x)/R`, and `L_(y)/R`.

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 an object consisting of a particle with mass `\(M\)` connected to two particles, each with mass `\(m\)`, via two variable-length, massless arms, each of length `\(s\)`. This object is restricted to move on the surface of a sphere with a radius `\(R\)`, where `\(s \ll R\)`. A local Cartesian coordinate system `\(XY\)` is established on the sphere. At time `\(t \leq 0\)`, the object is at position `\((X, Y) = (0, 0)\)` with both arms in a retracted state (`\(s=0\)`) and is at rest. When `\(t > 0\)`, the object begins to move, with both arms extending to the same length and symmetrically opposite in direction to the X-axis. Let the angle between the two arms be `\(2\theta\)`, then the relative position of one of the particles with mass `\(m\)` to the particle of mass `\(M\)` is `\((x,y)=s\times(\cos\theta ,\sin\theta )\), and the other \(s\times(\cos\theta, -\sin\theta)\)`. As the object moves, `\((x, y)\)` undergoes periodic changes as described:

`\((x,y)\colon(0,0)\to\left(0,L_{y}\right)\to\left(L_{x},L_{y}\right)\to\left(L_{x},0\right)\to(0,0)\)`

If \\((x,y)\\) to the next of the 4 states with frequency \\(f\\), find the (signed) average velocity `\(\langle V \rangle\)` of the object's motion, where positive \\(V\\) is in the direction of the positive \\(X\\) axis. Assuming `\(m \ll M\)` and `\(L_x, L_y \ll R\)`, in the calculations, you only need to take the non-zero lowest order terms of `\(m/M\)`, `\(L_x/R\)`, and `\(L_y/R\)`.

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.