A package of mass 5 kg sits at the equator of an airless asteroid of mass 3.0 x 1020 kg and radius 6.3 x 105 m. We want to launch the package in such a way that it will never come back, and when it is very far from the asteroid it will be traveling with speed 237 m/s. We have a large and powerful spring whose stiffness is 1.8 x 105 N/m. How much must we compress the spring? |compression| (a positive number) = m

Three dimensions. Three point particles are fixed in place in an xyz coordinate system. Particle A, at the origin, has mass ma. Particle B, at xyz coordinates (3.00d, 1.00d, 3.00d), has mass 4.00ma, and particle C, at coordinates (-2.00d, 1.00d, -1.00d), has mass 2.00ma. A fourth particle D, with mass 3.00ma, is to be placed near the other particles. If distance d = 4.50 m, at what (a) x, (b) y, and (c) z coordinate should D be placed so that the net gravitational force on A from B, C, and D is zero? (a) Number Units (b) Number |Units (c) Number Units

13.39 • CALC Consider the ring- shaped object in Fig. E13.39 D. A particle with mass m is placed a distance x from the center of the ring, along the line through the center of the ring and perpendicular to its plane. (a) Calculate the gravitational potential energy U of this system. Take the potential energy to be zero when the two objects are far apart. (b) Show that your answer to part (a) reduces to the expected result when x is much larger than the radius a of the ring. (c) Use F. = -dU/dx to find the magnitude and direction of the force on the particle (see Section 7.4 9). (d) Show that your answer to part (c) reduces to the expected result when x is much larger than a. (e) What are the values of U and F, when x = 0? Explain why these results make sense. Figure E13.39 т M

11. A freely falling solid body passes two measuring points that are 12 m below each other during a time of 1 s. From what height above the upper measuring point starts the free fall of the body? What is its speed at each of the two measuring points? [height 2.566 m; speeds 7.1 m/s and 16.9 m/s]