PHYSICS 103 FINAL EXAM

Transcription

1 19 Jan 08 Printed Name: Please Circle your Section! 9 am Jau 10 am Marino 9 am Garcia-Garcia 10 am Garcia-Garcia 9 am Ianni 10 am Ianni 9 am Rothman 10 am Rothman 9 am Yazdani 10 am Yazdani 9 am Yost 10 am Yost PHYSICS 103 FINAL EXAM Problem Score 1 /50 2 /50 3 /50 4 /50 5 /50 6 /50 Total /300 Instructions: When you are told to begin, check that this examination booklet contains all the numbered pages from 5 through 25. Princeton University Undergraduate Honor Committee This examination is administered under the Princeton University Honor Code. Students should sit one seat apart from each other, if possible, and refrain from talking to other students during the exam. All suspected violations of the Honor Code must be reported to the Honor Committee Chair at honor@princeton.edu. The checked items below are permitted for use on this examination. Any item that is not checked may not be used and should not be in your working space. Assume items not on this list are not allowed for use on this examination. Please place items you will not need out of view in your bag or under your working space at this time. University policy does not allow the use of electronic devices such as cell phones, PDAs, laptops, MP3 players, ipods, etc. during examinations. Students may not wear headphones during an examination. Course textbooks Course Notes Other printed materials NOTHING IS ALLOWED EXCEPT THIS EXAMINATION, A PEN OR PENCIL, AND YOUR CALCULATOR. THE CALCULATOR POLICY (SEE NEXT PAGE) IS IN PLACE AS ALWAYS. Students may only leave the examination room for a very brief period without the explicit permission of the instructor. The exam may not be taken outside of the examination room. This is a timed examination. You will have 3 hours to complete this exam. During the examination, the Professor or a preceptor will be outside the exam room. Rewrite and sign the pledge: I pledge my honor that I have not violated the Honor Code during this examination. Signature

2 FURTHER INSTRUCTIONS The exam contains SIX problems with multiple parts, worth varying numbers of points, which total to 300 points. Do not panic or be discouraged if you cannot do every part of every problem; there are both easy and hard parts in this exam. If a part of a problem depends on a previous answer you have not obtained, assume it and proceed. Keep moving and finish as much as you can! Read each problem carefully. You must show your work the grade you get depends on how well the grader can understand your solution even when you write down the correct answer. Always write down analytic answers first and only then calculate numerical values if requested. Check that the dimensions of your answers are correct. Please Box Your Answers. To refresh your memory, here is the calculator policy from the Blackboard website. Calculator Policy: Calculators are tools used by all physicists. Many of these devices have advanced features that are very useful. However, not all students have calculators with these features, and our rules are designed for fairness. On tests and the final: No advanced features may be used at any time. Calculator use is restricted to arithmetic and evaluation of trig functions, exponentials, logs, square roots, etc. You are not allowed to use, for example, calculator memory for storage of physics formulas or constants automatic equations solving graphing features functions such as quadratic formula You are also not allowed to use of any of these features for checking answers. Violation of these rules is a violation of the Honor Code. DO ALL THE WORK YOU WANT TO HAVE GRADED IN THIS EXAMINATION BOOKLET! YOU WILL NOT BE ALLOWED TO HAND IN ANYTHING ELSE.

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5 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 1, Page 5 Problem 1. Conservation Laws The spring in the figure has a spring constant k = N/m. It is compressed a distance x 0 = 15 cm, then launches a block of mass m = 0.20 kg. The horizontal surfaces and the inclined plane are frictionless. The second horizontal surface is a height h = 2.0 m above the first, and the angle of the inclined plane is θ = 45. a) (12 pts) For each question below, first answer yes or no, then write 1-2 sentences explaining your reasoning. i) Is the energy of the block conserved after it leaves the spring? ii) Is the momentum conserved after the block is released by the spring, while the block is on the lower horizontal surface? iii) Is the momentum conserved while the block is on the inclined plane? Comment on both the horizontal and vertical components of the momentum. iv) Is the momentum conserved while the block is sailing through the air? Comment on both the horizontal and vertical components of the momentum.

6 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 1, Page 6 b) (15 pts) What is the block speed v t at the top of the incline? What is the block speed v 2 just before landing? What distance d (see figure) on the upper horizontal surface does the block travel before landing? v t : v 2 : d:

7 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 1, Page 7 c) (10 pts) The frictionless surface on the incline is replaced by one such that the coeffcient of kinetic friction between the block and the surface is µ k = Find the work done on the block by the spring (W spr ) and by friction (W f ). What is the work W g,1 done by the gravitational force while the block is on the inclined plane? What is the work W g,2 done by the gravitational force while the block is flying? W spr : W f : W g,1 : W g,2 :

8 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 1, Page 8 d) (13 pts) What is the distance d in this case (with friction on the inclined plane)? d:

9 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 2, Page 9 Problem 2. Collisions and Kinematics a) (20 pts) A particle of mass m = 10.0 g moves with a velocity v 0 = 60 m/s along the x-axis. The particle hits a body of mass M = 2.0 kg and sticks to it. Determine the velocity v f of the body plus particle after the collision. Determine the amount of energy E lost in the collision. v f : E:

10 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 2, Page 10 b) A particle of mass m 1 =100 g moves along the x-axis with velocity v 0 = 2.0î m/s. The particle hits a second particle of mass m 2 = m 1, which is at rest. The collision is elastic. After the collision m 1 has velocity v 1 and m 2 has velocity v 2. i) (10 pts) Prove that v 1 v 2 = 0 in this case. (Hint: use conservation of momentum and energy, and the definition of the dot product.)

11 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 2, Page 11 ii) (20 pts) After the collision v 1 makes an angle θ 1 =45 with respect to v 0. Determine the angle θ 2 that v 2 makes to v 0, and find the magnitudes v 1 and v 2. θ 2 : v 1 : v 2 :

12 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 3, Page 12 Problem 3. Binary Star System Two stars each have mass M and radius R. One has its center on the origin of an xyz coordinate system, and the center of the other is at x 2 = 10R. A spacecraft of mass m (with m << M) moves along the x-axis under the force of gravity. (Its engines have been destroyed.) a) (12 pts) Below, sketch a graph of the space capsule s potential energy as it moves on the x-axis. Don t forget to label the axes. Indicate the value of the potential energy at any extrema, and the positions of any extrema, and how the potential energy varies with distance from each star when it is very near the star.

13 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 3, Page 13 b) (6 pts) Suppose the spacecraft is at rest exactly midway between the stars. Is this a point of equilibrium? If so, is it a stable or unstable equilibrium point? c) (4 pts) Suppose at t = 0, the spacecraft is at rest at x = 4R. What is the net force F on the spacecraft? F = Direction:

14 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 3, Page 14 d) Suppose instead that at t = 0, the spacecraft is at rest at x = 5R. The captain fires a huge cannonball of mass m c = m/10 in the y direction. After the cannon is fired, the spacecraft has net mass (9/10)m and moves off in the +y direction. i) (8 pts) Is angular momentum conserved before and after the cannonball is fired? If v c is the speed of the cannonball after the cannon is fired, what is the magnitude L c of the angular momentum of the cannonball about the z-axis? yes or no: L c :

15 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 3, Page 15 ii) (20 pts) With must v c be in order to permit the spacecraft to escape the gravitational pull of the binary star system? As always, show all the steps of your work clearly for full credit. v c :

16 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 4, Page 16 Θ Problem 4. Banked Curves An engineer designing the bank on a curved road has to take friction into account. Suppose the road is curved into a circle at a certain point. The radius of the circle is R (measured to the position of the car). Assume the driver wants to negotiate the curve while maintaining some speed v. The coefficient of friction between the wheels and the surface of the road is µ s. a) (15 pts) Draw a freebody diagram of the car. Label every force clearly. You do not need to resolve the forces on your diagram, but make it clear where the angle θ comes in.

17 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 4, Page 17 b) (20 pts) For what speed v is the angle θ a perfect bank? (On a perfect bank, the car is able to go around the curve, but the road exerts no sidewise frictional force on the car.) Express your answer for v in terms of θ, g, and R. v:

18 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 4, Page 18 c) (15 pts) At what speed ṽ will the car just begin to slide up a bank of angle θ? Express ṽ in terms of θ, g, R, and µ s. ṽ:

19 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 5, Page 19 Problem 5. Oscillations and Pulleys In the system shown in the figure, a spring with spring constant k is attached to a massless string which passes over a pulley and is attached to a mass m. The pulley is a homogeneous disk of mass m p and radius R. (It is supported by a rigid rod extending out from the table, but the details of that do not matter.) The disk rotates without friction around its axis perpendicular to the page. The string does not slip on the pulley. a) (10 pts) First, take m p =0. The mass is at rest in its equilibrium position when it is at y = 0 and the spring is elongated by d = m. Find the tension T 0 in the string when the mass is at y = 0. Express your answer in terms of variables given in the problem and g as needed. T 0 :

20 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 5, Page 20 b) (10 pts) Find the period P of oscillations of the mass m, assuming that the amplitude A of the oscillations satisfies A < d so that the string never goes slack. After finding an algebraic expression for the period, also provide a numerical answer in seconds. (Hint: yes, you have all the information you need to do so.) P, algebraic: P, numerical:

21 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 5, Page 21 c) (10 pts) What is the direction of the net torque on the pulley when the mass m is below the equilibrium position and moving downward? You must defend your answer with 1 or 2 sentences and a sketch for full credit. direction:

22 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 5, Page 22 d) (20 pts) Determine the period P of oscillations of the mass m in its motion along the y axis. As before, the spring is elongated by d = m in the equilibrium configuration, and you should consider only oscillations with amplitudes smaller than d. In this case, after finding an algebraic expression, use the following values to find a numerical result: m p =1.0 kg and m= 0.10 kg. The moment of inertia for the disk is I = (1/2)MR 2. P, algebraic: P, numerical:

23 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 6, Page 23 Problem 6. Spiral String Wave A string with tension T is found in the lab. Instead of vibrating the string, someone spirals it and produces a spiral wave, as sketched in the figure. A spiral wave on a string, traveling along the z axis, can be thought of as a superposition of two independent sinusoidal waves undulating along the two orthogonal directions, î and ĵ. Mathematically, this spiral wave is described by R(z, t) = R sin(kz ωt)î + R sin(kz ωt + φ)ĵ. Here, R is the wave amplitude, and φ = π/2. The values k and ω are constants. a) (5 pts) Based on the R(z, t) given above, in what direction is the wave traveling (+ˆk or ˆk)? direction:

24 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 6, Page 24 b) (5 pts) Consider the small chunk of string in the xy plane (that is, with z = 0). As the figure on the right indicates, this chunk is executing circular motion as the wave travels through. Since φ = +π/2, what is the direction of the circular motion (clockwise or counterclockwise)? direction: c) (15 pts) Consider that small chunk to have length δz, with δz small enough that you can treat the chunk as a point mass. What is its kinetic energy, K? What is its angular momentum L about the origin? Express your answers in terms of k, ω, T, R, and δz as needed.. (Hint: Remember that the velocity of the wave is T/µ, and note that the mass of the chunk is m = µδz. Don t leave µ in your answers.) K: L = Direction:

25 Physics 103 Last In-Class Exam, 18 Dec 07 Problem 6, Page 25 d) (15 pts) Considering that the chunk of string is executing circular motion, what is the net force F on that chunk of string? Express your answer in terms of k, ω, T, R, and δz as needed. F = Direction: e) (10 pts) You should have found in the previous part that the force F on the chunk of string is a function of R. Find the potential energy U of the chunk, taking the zero point to be at the origin. (Hint: you might need to evaluate an integral along a radius.) Express U in terms of k, ω, T, R, and δz. U:

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