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A.3 A particle of mass m is subject to a central force F towards the origin given by F = -mf(r) f where r is the position vector, f is its direction and r = F. a. Write down the equations of motion in plane polar coordinates where: ↑ = cos 0x + sin Øy Ô = -sin 0x + cos bý and use one of them to show that the quantity h = r²d is conserved. You may use the expression for the acceleration (no need to derive it): ď²ŕ dt² b. Using the equations of motion, derive a differential equation for the inverse distance u(0) = 1/r(t) as a function of the angle 0. c. In a frame S' rotating at angular frequency , the general relationship between a vector à in S' and an inertial frame S is given by: dA dA = +wx Ā dt dt S S' Show that an observer on Earth will attribute "fictitious" forces acting on an object of mass m and velocity Vs relative to the Earth given by Ffict = -mw x (x F) - 2mw X Vs Identify which term is the Centrifugal force and which is the Coriolis force. d. Show that, to first order in w, the velocity of the object in a rotating frame is v(t)s¹ = −gt2 + Vs (0) — × [−gt²2+2tvs (0)] Justify any approximations you make. You may assume the only acceleration in the problem is gravity, so as = -g2. e. Draw a diagram of the rotating Earth and identify the directions in which the Centrifugal and Coriolis forces act. If an object travels north from the equator, identify on the diagram the direction that the Coriolis force deflects the motion. (Note: The Earth rotates from west to east.) - [* - - ( )] + + + + + +