(6%) Problem 2: A hoop rolls down a 4.75 m high hill without slipping. Randomized Variables d=4.75m I (it, What is the nal speed of the hoop, in meters per second? (12%) Problem 5: The planet Earth orbits around the Sun and also spins on its own axis. [g 33% Part (a) Calculate the angular momentum, in kilogram meters squared per second, of the Earth in its orbit around the Sun. {3 33% Part (b) Calculate the angular momentum, in kilogram meters squared per second, of the Earth spining on its axis. A: 33% Part (c) How many times larger is the angular momentum of the Earth in its orbit than the angular momentum of the Earth spinning on its axis? (12%) Problem 6: Modern wind turbines generate electricity from wind power. The large, massive blades have a large moment of inertia and carry a great amount of angular momentum when rotating. A wind turbine has a total of 3 blades. Each blade has a mass of m = 5500 kg distributed uniformly along its length and extends a distance r = 49 m from the center of rotation. The turbine rotates with a frequency of f = 11 rpm. Q 25% Part (a) Enter an expression for the total moment of inertia of the wind turbine about its axis of rotation, in terms of the dened quantities. Q 25% Part (b) Calculate the total moment of inertia of the wind turbine about its axis, in units of kilogram meters squared. Q 25% Part (c) Enter an expression for the angular momentum of the wind turbine, in terms of the dened quantities. Q 25% Part (d) Calculate the angular momentum of the wind turbine, in units of kilogram meters squared per second. (6%) Problem 7: A playground merry-go-round with a mass of 125 kg and a radius of 1.9 m is rotating with a frequency of 0.48 rev/s. Randomized Variables m1 = 125 kg m2 = 27 kg f1 = 0.48 rev/s r = 1.9 m [A What is the magnitude of its angular velocity, in radians per second, after a 27 kg Child gets onto it by grabbing its outer edge? The child is initially at rest. (6%) Problem 8: On the surface of the Moon an astronaut has a weight of F g = 150 N. The radius of the Moon is Rm = 174 x 106 In, the gravitational constant is G = 6467 x 10-11 N (m/kg)2 and the mass of the Moon is M = 7.35 x 1022 kg. & Calculate the mass of the astronaut, m, in kilograms. (6%) Problem 9: A satellite m = 500 kg orbits the earth at a distance d = 229 km, above the surface of the planet. The radius of the earth is re = 6.38 x 106 In and the gravitational constant G = 6.67 x 10'11 N rn2/1(g2 and the Earth's mass is me = 5498 x 1024 kg. & What is the speed of the satellite in m/s? (6%) Problem 10: Astrology, that unlikely and vague pseudoscience, makes much of the position of the planets at the moment of one's birth. The only known force that a planet could exerts on us is gravitational, so if there is anything to astrology we should expect this force to be signicant. 65 33% Part (3) Calculate the gravitational force, in newtons, exerted on a 3.7 kg baby by a 105 kg father who is a distance of 0.075 m away at the time of its birth. Q 33% Part (b) Calculate the force on the baby, in newtons, due to Jupiter (the largest planet, which has a mass of 1.90 X 1027 kg) if it is at its closest distance to Earth, 6.29 X 1011 m away. 43 33% Part (c) What is the ratio of the force of the father on the baby to the force of Jupiter on the baby? (6%) Problem 11: The strength of the gravitational eld of a source mass can be measured by the magnitude of the acceleration due to gravity at a eld point. Thus the gravitational eld strength at a point depends on the distance from the source mass. For this problem assume that Earth's mass is concentrated at its center and that Earth has a radius of RE = 6,378 km at sea level and a mass of ME = 5.972x1024 kg. Ignore any forces due to Earth's rotation or due to other astronomical bodies. Q 33% Part (a) Enter an expression for the magnitude of the acceleration due to gravity near Earth, in terms of RE, ME, the altitude above the Earth's surface, H, and the gravitational constant, G. Q 33% Part (b) Calculate the magnitude of the acceleration due to gravity, in meters per second squared, at sea level on Earth. Q 33% Part (c) Calculate the magnitude of the acceleration due to gravity, in meters per second squared, near Earth at an altitude of H = 780 km above sea level. (6%) Problem 12: A moon of mass m and radius a is orbiting a planet of mass M and of radius b at a distance d (center-to-center) in a circular orbit. A 50% Part (a) Derive an expression for the speed of the moon v in terms of M, d and the gravitational constant G. A 50% Part (b) Derive an expression for the total mechanical energy E of the moon in terms of m, M, d and the gravitational constant G. (6%) Problem 13: An object of mass m is launched from a planet of mass M and radius R. 45 50% Part (a) Derive and enter an expression for the minimum launch speed needed for the object to escape gravity, Lei to be able to just reach r = 00. [A 50% Part (b) Calculate this minimum launch speed (called the escape speed), in meters per second, for a planet of mass M = 8.88 X 1024 kg and R = 872 x102 km