Two astronauts, each having a mass of 95.5 kg, are connected by a 10.0-m rope of negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed of 4.60 m/s. Treating the astronauts as particles, calculate each of the following. CG × d (a) the magnitude of the angular momentum of the system kg m2/s (b) the rotational energy of the system KJ By pulling on the rope, the astronauts shorten the distance between them to 5.00 m. (c) What is the new angular momentum of the system? kg m2/s (d) What are their new speeds? m/s (e) What is the new rotational energy of the system? KJ (f) How much work is done by the astronauts in shortening the rope? KJ

A uniform horizontal disk of radius 5.50 m turns without friction at w = 2.55 rev/s on a vertical axis through its center, as in the figure below. A feedback mechanism senses the angular speed of the disk, and a drive motor at A ensures that the angular speed remain constant while a m = 1.20 kg block on top of the disk slides outward in a radial slot. The block starts at the center of the disk at time t = 0 and moves outward with constant speed v = 1.25 cm/s relative to the disk until it reaches the edge at t = 360 s. The sliding block experiences no friction. Its motion is constrained to have constant radial speed by a brake at B, producing tension in a light string tied to the block. (a) Find the torque as a function of time that the drive motor must provide while the block is sliding. Hint: The torque is given by t = 2mrvw. t N.m (b) Find the value of this torque at t = 360 s, just before the sliding block finishes its motion. N.m (c) Find the power which the drive motor must…

(a) A planet is in an elliptical orbit around a distant star. At its closest approach, the planet is 0.670 AU from the star and has a speed of 54.0 km/s. When the planet is at its farthest distance from the star of 36.0 AU, what is its speed (in km/s)? (1 AU is the average distance from the Earth to the Sun and is equal to 1.496 × 1011 m. You may assume that other planets and smaller objects in the star system exert negligible forces on the planet.) km/s (b) What If? A comet is in a highly elliptical orbit around the same star. The comet's greatest distance from the star is 25,700 times larger than its closest distance to the star. The comet's speed at its greatest distance is 2.40 x 10-2 km/s. What is the speed (in km/s) of the comet at its closest approach? km/s