You are deep in outer space (far from any gravitational sources) and standing at the center of a rotating platform (it lies in the plane z=0, and rotates ccw with angular velocity 2). you look down and see a puck on the platform with coordinates (x, y) (in your, the tt, frame). a) write down newton's 2nd law for the x & y coordinates of the puck as you observe it (2 eqs) b) write down the general solution to these two equations by changing to complex coordinates (i. e. write n = x + iy) and guessing that one of the solutions has the form n = e-iat (what is the form of the second solution, when you have two solutions which have the same form? this arose in taylor's discussion of the critically damped oscillator in section 5.4) c) at time t=0 the puck is launced from position to = (x0,0) with velocity v = (vxo, vyo) (all in your frame). show that the time dependent coordinates obey: x(t) = (xo + vxot) cos(at) + (vyo + nx)t sin(nt) y(t) = -(xo + vxot) sin(t) + (vvo + 1x)t cos(at)

Amass m = 74 kg slides on a frictionless track that has a drop, followed by a loop-the-loop with radius r = 18.4 m and finally a flat straight section at the same height as the center of the loop (18.4 m off the ground). since the mass would not make it around the loop if released from the height of the top of the loop (do you know why? ) it must be released above the top of the loop-the-loop height. (assume the mass never leaves the smooth track at any point on its path.) 1. what is the minimum speed the block must have at the top of the loop to make it around the loop-the-loop without leaving the track? 2. what height above the ground must the mass begin to make it around the loop-the-loop? 3. if the mass has just enough speed to make it around the loop without leaving the track, what will its speed be at the bottom of the loop? 4. if the mass has just enough speed to make it around the loop without leaving the track, what is its speed at the final flat level (18.4 m off the ground)? 5. now a spring with spring constant k = 15600 n/m is used on the final flat surface to stop the mass. how far does the spring compress?