PHY 2121/2521 ASSIGNMENT 2 OF 2022 DUE DATE:06 MAY 2022 1. A primed coordinate system rotates about the 𝑧-axis with angular velocity 𝜔 ⃗ = 𝑖̂′ cos 𝑡 + 𝑗̂′ sin 𝑡 relative to a fixed coordinate system, where 𝑡 is the time. The origin of the primed system has position vector 𝑅⃗0 = 𝑖̂′ 𝑡 − 𝑗̂′ + 𝑘̂ ′ 𝑡 2 with respect to the fixed system. The position vector of a particle is given by 𝑟 ′ = 𝑗̂′ 𝑡 + 𝑘̂ ′ relative to the moving system. Calculate the true velocity of the particle and the acceleration of the moving system. 2. A particle moving in a central force field located at 𝑟 = 0 describes the spiral 𝑟 = 𝑒 −𝜃 . Show that the force varies as the inverse third power of 𝑟. 3. Using 𝑟 = 𝑎(1 − cos 𝜃) in the equation 𝑓(𝑟) = 𝑚𝑙 2 𝑑 2 𝑟 2 𝑑𝑟 2 [ − ( ) − 𝑟] 𝑟 4 𝑑𝜃 2 𝑟 𝑑𝜃 show that the force varies as the inverse fourth power of 𝑟. 4. Show that the position of a particle as a function of time 𝑡 can be determined from the equation 𝑡 = ∫[𝐺(𝑟)]−1/2 𝑑𝑟, where 𝐺(𝑟) = 2𝐸 2 𝑙2 + ∫ 𝑓(𝑟) 𝑑𝑟 − 2 𝑚 𝑚 𝑟 END OF ASSIGNMENT