First Name: ____________________ Last Name:_____________________ Section: ________ ■ 1 April 23, 2008 Physics 207 EXAM 3 Please print your name and section number (or TA’s name) clearly on the first page. Show all your work in the space immediately below each problem. Your final answer must be placed in the boxes, when provided. Problems will be graded on reasoning and intermediate steps as well as on the final answer. Be sure to include units wherever necessary, and the direction of vectors. Each problem is worth 20 points. Try to be neat! Check your answers to see that they have the correct dimensions (units) and are the right order of magnitude. You are allowed one sheet of notes (8.5” x 11”, 2 sides), a calculator, and the constants in this exam booklet. The exam lasts exactly 90 minutes. Constants: Acceleration due to gravity at the earth’s surface: g = 9.81 m/s2 SCORE: Problem 1: __________ Problem 2: __________ Problem 3: __________ Problem 4: __________ Problem 5: __________ TOTAL: ___________ Don't open the exam until you are instructed to start. "Whatever you dream you can do begin it. Boldness has genius, power and magic in it. Begin it now."………….GoetheFirst Name: ____________________ Last Name:_____________________ Section: ________ ■ 2 Problem 1 A uniform rectangular beam of length L= 5 m and mass M=40 kg is supported, but not attached, to two posts which are length D=3 m apart. A child of mass W=20 kg starts walking along the beam. a) Assuming infinitely rigid posts, how close can the child get to the right end of the beam without it falling over? [Hint: The upward force exerted by the beam on the left cannot be negative – this is the limiting condition on how far the child can be to the right. Set up the static equilibrium condition with the pivot about the left end of the beam.] (10 pts)First Name: ____________________ Last Name:_____________________ Section: ________ ■ 3 b) Suppose the left end of the beam is attached to the left post although it can still freely pivot about it. The right post is not attached, as in part a). If the left post is infinitely rigid while the Young’s modulus for the right post, which is of length s=10 m, is Y=1010 N/m2 and its cross sectional area is A=1/2 m2, how much does its length differ between the situation when the child is not present and when the child is at the rightmost edge of the beam? (10 pts)First Name: ____________________ Last Name:_____________________ Section: ________ ■ 4 Problem 2 A child of mass M is on a swing whose seat and rope have negligible mass, and the rope has length L. Suppose the child can be treated as a point mass, and the rope makes an angle as a function of time θ(t) with respect to the vertical. a) Neglecting any friction in the system and assuming θ(t) is much smaller than unity and when there is nobody pushing the child, the differential equation satisfied by θ(t) is What is B in terms of g and L? (5 pts)First Name: ____________________ Last Name:_____________________ Section: ________ ■ 5 b) Solve the equation for θ(t) if are given as initial conditions (where B and C are positive numbers). (5 pts) c) For the motion given in part b), what is the child’s maximum kinetic energy? (5 pts)First Name: ____________________ Last Name:_____________________ Section: ________ ■ 6 d) Suppose an adult can push the child in the direction of the child’s motion just when the child passes through θ=0. To achieve maximum height for the child, approximately how many times should the adult push the child during a time period Q which is much longer than the natural oscillation period of the swing? [Hint: Think resonance.] (5 pts)First Name: ____________________ Last Name:_____________________ Section: ________ ■ 7 Problem 3 a.) Suppose we have a massive rope attached to the ceiling. The bottom end of the rope dangles in midair. The mass of the rope is 2.7 kg and its length is 4.5 meters. The bottom end of the rope is shaken and then released to produce a pulse that travels up the rope. What is the speed of the pulse when it is halfway up the rope? (6 pts) Here’s a new question. b.) The figure shows a snapshot graph at t = 0 s of a sinusoidal wave traveling to the right along a string at 50 m/s. Write the equation that describe the displacement D(x, t) of this wave. Your equation should have numerical values, including units, for all quantities except x and t. (8 pts)First Name: ____________________ Last Name:_____________________ Section: ________ ■ 8 c.) What is the maximum acceleration of any portion of the string described in part b ? (6 pts)First Name: ____________________ Last Name:_____________________ Section: ________ ■ 9 Problem 4 Consider a string of length L that is fixed at one end and free at the other end. a.) Sketch the standing wave patterns for the lowest three resonant frequencies of the string. (Note: there will always be a node at the fixed end and an antinode at the free end.) ( 3 pts) b.) These three modes correspond to wavelengths λ1, λ2, and λ3. Express λ1, λ2, and λ3 in terms of L and numerical factors. (3 pts) c.) Write an expression for the wavelength λn of the “nth” resonant mode in terms of L, n, and numerical factors. (3 pts) d.) Given that the speed of the waves is v, what are the three lowest resonant frequencies, f1, f2, and f3 ? (3 pts)First Name: ____________________ Last Name:_____________________ Section: ________ ■ 10 Here’s a new question: e.) Two loudspeakers are placed 1.8 m apart. They play tones of equal frequency. If you stand 3.0 m in front of the speakers, and exactly between them, you hear a maximum of intensity. As you walk parallel to the plane of the speakers, staying 3.0 m away, the sound intensity decreases until reaching a minimum when you are directly in front of one of the speakers. The speed of sound in the room is 340 m/s. What is the frequency of the sound? (8 pts)First Name: ____________________ Last Name:_____________________ Section: ________ ■ 11 Problem 5 A ball hangs from a balance. When the ball hangs in the air, the balance reads 0.1 N. When the ball is lowered into water, the balance reads
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