9.2 A donut-shaped space station (outer radius R) arranges for artificial gravity by spinning on the axis of the donut with angular velocity w. Sketch the forces on, and accelerations of, an astronaut standing in the station (a) as seen from an inertial frame outside the station and (b) as seen in the astronaut's personal rest frame (which has a centripetal acceleration A = w²R as seen in the inertial Problems for Chapter 9 361 frame). What angular velocity is needed if R = 40 meters and the apparent gravity is to equal the usual value of about 10 m/s²? (c) What is the percentage difference between the perceived g at a six-foot astronaut's feet (R = 40 m) and at his head (R = 38 m)?

Around 2.5 centuries ago, several physicists of the time came up with the notion of a dark star.  This was a star so dense, with so much gravity, that not even light could escape.  The calculations used Newtonian mechanics.  In class, we calculated the escape speed from the surface of the earth or the distance from the sun, and the mass of the planet or star. Here, the process is partially reversed.  Calculate the dark star radius from the mass of the star and the escape speed.  Answer in kilometers. c = 3*108 m/s M = 3.2*1030 kg G = 2/3 * 10-10 N*m2/kg2

Problem 1. Consider the Sun-Earth system with a center-to-center distance of 1.5x 101 m. Suppose that at some instance the Sun's velocity is zero and its location is at the origin. Ignoring all effects but that of the Earth, what will the Sun's velocity and position be after 1 day. Compute the same quantities for the Earth ignoring the fact the Earth is in a circular orbit (i.e., assume it to initially be at rest). Treat this as a 1D problem. Gr Ecrok FJ 2/A 2 57 (5.994 Xt0 49 5,21 10 Problem 2. Given only the distance between the Earth and Moon (REM = 3.84 x 108 m) and that between the Earth and the Sun (1 AU), determine the mass of the Earth and the mass of the Sun. How can we measure REM or RSE? M 1 142 1 k (P

A bead of mass m slides without friction along a curved wire with shape z = f(r) where r = Vr2 + y², i.e. the distance from the z-axis. The wire is rotated around the z-axis at a constant angular velocity w. Gravity acts downward along the z-axis with a constant acceleration g. a) Using Newton's second law in an inertial frame, derive an expression for radius ro of a fixed circular orbit (i.e. a solution with r = ro = const.). What is the normal force the wire applies to the bead to keep it in a circular orbit? b) Show that the equation of motion for r(t) (general equation not the circular motion) is F(1 + f'(r)²) + i² f'(r)f"(r) + gf'(r) – w?r = 0. Using this verify your answer to part (a). c) Consider small displacements from the circular orbit, r = ro+e(t). Derive a condition on the function f(r) such that a circular orbit at r = ro is stable. d) Find the force on the bead in the o direction, i.e. perpendicular to the plane of wire. The angular velocity is w = . Obtain the answer…