Sun Earth Moon Mars Mass 1.99 10 30 kg 5.97 10 24 kg 7.35 10 22 kg 6.42 10 23 kg Radius 6.96 10 8 m 6.38 10 6 m 1.74 10 6 m 3.40 10 6 m Orbital Radius - 1.50 10 11 m 3.84 10 8 m 2.28 10 11 m Orbital Period - 3.15 10 7 s 27.3 days 687 days 2
1. (20 points) You do not need to show any work. No credit for blanks. One point each. 1. Sir Isaac Newton was in a hurry to get to the math battle, so he and his horse had to wait near the end of the rough drawbridge, before crossing the moat and proceeding to defend the virtue of calculus. For each of the following free-body diagrams, indicate whether (True) or not (False) the given drawing correctly reproduces all the forces acting on the draw-bridge. A. True False C. True False B. True False C. True False 2. Consider a small planet in an elliptical orbit having semi-major axis of a and eccentricity e (not equal to zero) around a massive star. Answer T for true or F for false. a. The distance between the planet and the star is 2a. b. The star is at one focus and the planet is at the other. c. The planet s kinetic energy is greater at perigee than at apogee. d. The planet s potential energy is greater at perigee than at apogee. e. The planet s total energy is greater at perigee than at apogee. f. The planet s angular momentum is greater at perigee than at apogee. 3
3. As shown in the diagram, a force is applied to a point on the left side of a crate, below the midpoint, and at an angle slightly above horizontal. The crate is neither tipping nor sliding. For each pair of [force and axis-of-rotation], characterize the torque produced by the force about that axis of rotation as either positive, negative, or zero. F X CM X O Force Axis of Rotation Circle +, 0, or - for each torque F g (force of gravity) O + 0 - F g CM + 0 - F O + 0 - F CM + 0 - F s (force of friction) O + 0 - F s CM + 0-4. Two masses M are fixed in place on the x-axis at x = ±L. A third mass m can be at different (x,y) locations. Answer T for true or F for false. a. Mass m has greater potential energy when at (0, L) than at (0, 0). b. Mass m has greater potential energy when at (0, L) than at (0, -L). c. Mass m has greater potential energy when at (0, 0) than at (0.999L, 0). d. Mass m has the greatest potential energy when infinitely far away from both M. 4 20
2. (20 points) A small block of mass m = 0.250 kg is attached to a string passing through a hole in a frictionless, horizontal surface. The block is originally revolving in a circle of radius r 1 = 0.800 m about the hole with a tangential speed of v 1 = 4.00 m/s. The string is then pulled slowly from below, shortening the radius of the circle in which the block revolves to r 2 = 0.25 m at which point the string breaks. What was the tension in the string when it broke? 5 20
3. (10 points) A different (and rather dangerous) kind of gyroscope is made using a slender rod in place of the usual circular disc. The rod is L = 0.200 m long and has a mass of M = 0.120 kg. The axle goes through the middle of the rod, has negligible mass, and is 0.08 m long (4 cm on either side of the rod). When the rod is made to spin at 150 radians per second and supported from one end of the axle, how long does it take to precess one full turn? 6 10
4. (18 points) A uniform, cube-shaped box on variable angle ramp with rough surface (m s = 1.1). As the ramp is raised from horizontal, will the box tip over or begin sliding first? For full credit, you must prove it by finding both the angle at which it will tip over q tip and the angle at which it begins to slide q slide. Draw a free-body diagram for each case. 7 18
5. (20 points) At a certain point, an asteroid is located exactly equidistant from both the Earth and the Moon, such that the three of them form an equilateral triangle. Set-up your coordinates as follows: place Earth at the origin, the Moon on the +x axis, and the satellite in the first quadrant (+x and +y). Find the total gravitational force acting on the asteroid, F. TOT Use the astronomical data provided on page????. Take the asteroid s mass to be M A = 8.67 10 17 kg, and use the obvious labels M E and M M. 9 20
6. (12 points) A child's toy consists of a spherical object of mass m = 0.050 kg attached to a spring. One end of the spring is fixed to the side of the baby's crib so that when the baby pulls on the toy and lets go, the object oscillates horizontally with a simple harmonic motion. The amplitude of the oscillation is 0.060 m and the maximum velocity achieved by the toy is 3.2 m/s. What is the kinetic energy K of the toy when the spring is compressed 0.042 m from its equilibrium position? 10 12