Physics 53 Summer 2008 1 Final Exam Solutions In questions or problems not requiring numerical answers, express the answers in terms of the symbols given, and standard constants such as g. If numbers are required, use g = 10 m/s2. Part A: Multiple choice questions. Check the best answer. Each question has a value of 4 points. 1. Which is a correct statement of a fundamental law of mechanics? If all the external forces on a system are conservative, total mechanical energy is conserved. If the total external force on a system is zero the total angular momentum about any point is conserved. If only conservative forces do work, the total linear momentum of the system is conserved. √ If any component of the total external force is zero, that component of the linear momentum is conserved. 2. In simple harmonic motion in one dimension: √ The acceleration and the displacement are always in opposite directions. The acceleration and the velocity are always in opposite directions. The velocity and the acceleration have their greatest magnitudes at the same time. None of the above is true. 3. A billiard ball (a solid sphere of radius R and mass m) moves to the right across a rough table. It started without rotating, so it is still skidding, although eventually it will roll without slipping. Which of the following is NOT true? The torque about its CM is clockwise. √ Both its CM speed and its angular speed of rotation are decreasing. [Angular speed is increasing] It angular momentum about a point on the table is constant. Its total kinetic energy is decreasing.Physics 53 Summer 2008 2 4. Three objects (1,2,3) start from rest at the same time from the same height on an incline, and roll to the bottom without slipping. Their moments of inertia obey I1> I2> I3. The order of speeds with which they will reach the bottom is: 1,2,3 √ 3,2,1 1,3,2 2,3,1 5. Shown is an auto with the normal forces on the front and rear wheels indicated. When the auto is at rest N1 is somewhat larger than N2 because the CM of the auto is closer to the front wheels. When the auto accelerates forward: Both normal forces remain the same. Both normal forces increase. N1 increases while N2 decreases. √ N2 increases while N1 decreases. [Consider torques about the CM.] 6. Concerning Kepler’s three laws, which of the following is NOT true? One law can be used to determine the mass of a planet if it has a moon. √ One law says that the quantity T2/ a3 (T is the period, a the semi-major axis) is the same for the earth’s orbit around the sun and moon’s orbit around the earth. [Must orbit the same object] One law says the planetary orbits are ellipses with the sun at a focus. One law is equivalent to conservation of angular momentum of the orbit. 7. The change in internal energy of an ideal gas is given by !U = ncV!T: Only for processes at constant volume. Only for adiabatic processes. Only for isothermal processes. √ For any process. N1 N2Physics 53 Summer 2008 3 8. A heat engine operates between reservoirs at 300 K and 250 K. In each cycle it takes heat 300 J from the hot reservoir. The net entropy change of the engine itself is 1 J/K per cycle. [Zero] The engine’s efficiency is 1/5. [No greater than 1/6] √ At least 250 J of heat must be expelled to the cold reservoir. None of the above is true. Part B: True-false questions. Check the correct answer. Each question has a value of 3 points. 1. A particle acted upon only by a conservative force and released from rest will move toward lower potential energy. √ True False 2. When you unscrew the top of a bottle, your thumb and fingers exert forces in opposite directions but torques in the same direction. √ True False 3. Galileo’s finding, that an object dropped from rest in a gravitational field has an acceleration independent of its mass, is only valid as an approximation for objects near the earth’s surface. True √ False [Always exactly true] 4. Spring tides, when the sun and moon combine to make especially high tides, occur only once per lunar month, when the sun and moon are on the same side of the earth. True √ False [Twice, also when they are on opposite sides] 5. Nights in the desert can be quite cold, even though the days are very hot, because the clear sky allows rapid energy loss by radiation at night. √ True False 6. Any kind of heat flow from higher to lower temperature is an irreversible process. √ True FalsePhysics 53 Summer 2008 4 Part C: Problems. Work problems in the space provided, indicating your method clearly. Problems have the point values shown. 1. A hunter with a blow-gun intends to shoot a monkey hanging from a tree limb at height h above the ground. The hunter is at distance 3h from the base of the tree. Having studied physics, the hunter knows that if he aims his blow gun directly at the monkey, and shoots at the instant the monkey releases its grip on the limb to drop to the ground, then the dart will strike the falling monkey. But the dart must travel fast enough to hit the monkey before it reaches the ground. a. How long does it take the monkey to reach the ground? b. What must be the minimum horizontal component of the dart’s initial velocity? c. What must be the minimum initial speed of the dart? Give all answers in terms of h and g. [Draw a picture.] [15 points] a. The time is given by h =12gt2, so t = 2h/ g. b. It must go 3h in this time or less, so 3h = v0xt, or v0x= 3h ! g/2h = 3 gh/2. c. To shoot at the monkey, we must have v0y/v0x= h/3h = 1/3, so v0y=13v0x= gh/2. The speed is given by v2= v0x2+ v0y2= 9gh/ 2 + gh/ 2 = 5gh, or v = 5gh. h 3hPhysics 53 Summer 2008 5 2. A ball of mass m attached to a massless string of length R is moving as shown in a vertical circle. We are interested in the situation at the three points indicated. a. The ball has the minimum speed necessary to stay in the circle at point A. What is that speed? Explain. b. What is its speed at point B? c. Suppose when it reaches point C the string breaks. To what height h above C will it rise before it falls again? Give answers in terms of R and g. [15 points] a. The minimum downward acceleration is g (when the tension in the string is zero)