A reference frame appropriate for a location on the Earth's surface has unit basis vectors: (pointing one unit in the direction East at the given location) (pointing one unit in the direction North at the given location) ii (pointing one unit in the direction Up at the given location) a) In the above diagram, the z axis runs straight through the Earth, connecting the North and South poles. A plane has been drawn, upon which lie: Earth's centre, its poles, and a location P on its surface (at latitude 99). The unit vector g lies on this plane (pointing from Earth's centre to the equator). Not pictured is h, which is perpendicular to this plane, pointing directly into the page. Express the unit vectors f1, 2, and ii in terms of the vectors g, h and It b) Since the Earth is spinning, the direction of these vectors change (with respect to the xed stars) over time. Relative to the xed stars, the Earth's angular velocity is w = 297%: radians per day. A person standing on the Earth at P observes an object of mass m to be moving with velocity '7 = cit + y. Determine the coriolis force on this object. Express your answer in terms of vectors A n, , and f1. Suppose a long pendulum of length 3 is suspended from the ceiling, with a heavy bob of mass m. The pendulum is free to oscillate in any vertical plane (see diagram). Using the small angle approximation of a pendulum, show that the equations of motions governing this system are: a = 2ay 92:1: 1} = 2a;i: 92y, where Q = \"9/! 51 and a = 2x sin (,9 radians per day. (Hint: You may neglect the friction and the centrifugal force. Since the bob under goes only small oscillations about the equilibrium position, you can also assume all motion takes place in the horizontal plane.) By making the complex variable substitution 2: = :1: +iy, show that this system can be expressed as a single ODE 5 + 2m: + 922 = 0. Solve this equation. If the pendulum's initial position and velocity are given by; 3(0) 13(0) y(0) = 0 9(0) = v. 0: 0: nd 33(t) and y(t). Interpret this behaviour