1. In the figure, a square of edge length 24.0 cm is formed by four spheres of masses m1 = 5.30 g, m2 = 2.50 g, m3 = 0.500 g, and m4 = 5.30 g. In unit-vector notation, what is the net gravitational force from them on a central sphere with mass m5 = 2.10 g?
2. Two dimensions. In the figure, three point particles are fixed in place in anxy plane. Particle A has mass mA = 5 g, particle B has mass 2.00mA, and particle C has mass 3.00mA. A fourth particle D, with mass 4.00mA, is to be placed near the other three particles. What (a) x coordinate and (b) y coordinate should particle D be placed so that the net gravitational force on particle A from particles B, C, and D is zero (d = 18 cm)?
3. Three point particles are fixed in position in an xy plane. Two of them, particle A of mass 5 g and particle B of mass 11 g, are shown in the figure with a separation of dAB = 0.466 m at angle Θ = 30°. Particle C, with mass 8 g, is not shown. The net gravitational force acting on particle A due to particles B and C is 2.26 x 10-14 N at an angle of -163.8°. from the positive x axis. What are (a) the x coordinate and (b) the y coordinate of particle C?
4. Certain neutron stars (extremely dense stars) are believed to be rotating at about 0.93 rev/s. If such a star has a radius of 43 km, what must be its minimum mass so that material on its surface remains in place during the rapid rotation?
5. Two concentric spherical shells with uniformly distributed masses M1 and M2 are situated as shown in the figure. Find the magnitude of the net gravitational force on a particle of mass m, due to the shells, when the particle is located at each of the radial distances shown in the figure: (a) a, outside both shells, (b) b, between the two shells and (c) c, inside both shells. State your answers in terms of the given variables, using G when applicable.
6. The figure gives the potential energy function U(r) of a projectile, plotted outward from the surface of a planet of radius Rs. If the projectile is launched radially outward from the surface with a mechanical energy of -2.0 x 109 J, what are (a) its kinetic energy at radius r = 1.25Rs and (b) its turning point (see Module 8-3) if Rs = 4800 km?
7. Zero, a hypothetical planet, has a mass of 4.2 x 1023 kg, a radius of 3.3 x 106 m, and no atmosphere. A 10 kg space probe is to be launched vertically from its surface. (a)If the probe is launched with an initial kinetic energy of 5.0 x 107 J, what will be its kinetic energy when it is 4.0 x 106 m from the center of Zero? (b) If the probe is to achieve a maximum distance of 8.0 x 106 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?
8. In deep space, sphere A of mass 140 kg is located at the origin of an x axis and sphere B of mass 77 kg is located on the axis at x = 3.0 m. Sphere B is released from rest while sphere A is held at the origin. (a) What is the gravitational potential energy of the two-sphere system just as B is released? (b) What is the kinetic energy of B when it has moved 1.8 m toward A?
9. Flying Circus of Physics
An orbiting satellite stays over a certain spot on the equator of (rotating) Earth. What is the altitude of the orbit (called a geosynchronous orbit)?
10. Several planets (Jupiter, Saturn, Uranus) are encircled by rings, perhaps composed of material that failed to form a satellite. In addition, many galaxies contain ring-like structures. Consider a homogeneous thin ring of mass 3.7 x 1022 kg and outer radius 4.1 x 108 m (the figure). (a) What gravitational attraction does it exert on a particle of mass 84 kg located on the ring's central axis a distance 3.1 x 108 m from the ring center? (b) Suppose that, starting at that point, the particle falls from rest as a result of the attraction of the ring of matter. What is the speed with which it passes through the center of the ring?