Consider the following spacetime: ds² - (½ 7 - M) di² dr² dt² + +1² do². -M where > 0 and M > 0 are constants. (a) Find the effective potential governing the trajectories of timelike and null geodesics. (b) Sketch the effective potential and describe the possible trajectories of the radial null geodesics. (c) Does this spacetime contain a black hole? Reason your answer. (d) Find coordinates such that the metric (2) is regular at r = M.

The graph shows the displacement from equilibrium of a mass-spring system as a function of time after the vertically hanging system was set in motion at time t = 0. Assume that the units of time are seconds, and the units of displacement are centimeters. The first t-intercept is (0.75, 0) and the first maximum has coordinates (1.75, 2). (a) What is the period T of the periodic motion? T = 4 seconds (b) What is the frequency f in Hertz? What is the angular frequency w in radians / second? f =.25 Hertz W = 1.57 radians / second (d) Determine the amplitude A and the phase angle y (in radians), and express the displacement in the form y(t) = A cos(wt - Y), with y in meters. y(t) = 0.04cos(pit-0.75pl) meters (e) With what initial displacement y(0) and initial velocity y' (0) was the system set into motion? y(0) = meters y'(0) = meters / second

A steel ball weighing 128 pounds is suspended from a spring. This stretches the spring 128/257 feet. The ball is started in motion from the equilibrium position with a downward velocity of 9 feet per second.The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second) . Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t. (Note that this means that the positive direction for y is down.)

Find the stationary points of the autonomous equation x˙ = x2 − 4x4. Classify each stationary point as stable, unstable or semistable.