## COMP 4440/5440 – Dr. Erdemir Mobile Robotics Project (DUE 12/02/2015) HONOR CODE I pledge my honor that I have neither given nor received aid on this work. Do not sign until after you have completed your assignment. Name: Signature: 1. (Prerequisite) Given the asset package (it is in mytsu under assignments folder) download it, open a new project (don’t double click on the file, it won’t open), go to Assets/Import Package/Custom Package, select the asset package given to you (project.unitypackage). After the import is completed, you will see main scene in the assets folder of your project. Double click on it, and choose “Don’t Save” option if it asks for save. 2. Print this page and attach it to your code and your snapshot of your final scene (5 points) 3. After the class starts, instructor will come next to you and you are supposed to show him your code running (10 points) 4. When you run the code you will see, your robot (red cube that we used in the class) is trying to reach the targets but it can’t due to an obstacle between your robot and the targets. Write a code that makes this robot to avoid from the obstacles and reach the targets. Your code will read the collision and intelligently avoid from other moving robots and fixed obstacles and get the three targets. (50 points) 5. Maximum time is 2 minutes, your robot shall get the targets in 2 minutes (10 points) 6. Your robot shall escape from the blue robot and not collide them. (10 Points) 7. Anything extra (up to 20 points) ? Moving objects, new sensor, Artificial Intelligence or other techniques. 8. YOU CAN NOT USE TRANSLATE FUNCTION. USE ONLY AddRelativeForce FUNCTION IN THE FORWARD DIRECTION AS ALL THE MOBILE ROBOTS WORK.

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## Extra Credit Due: 11:59pm on Thursday, May 15, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A Man Running to Catch a Bus A man is running at speed (much less than the speed of light) to catch a bus already at a stop. At , when he is a distance from the door to the bus, the bus starts moving with the positive acceleration . Use a coordinate system with at the door of the stopped bus. Part A What is , the position of the man as a function of time? Answer symbolically in terms of the variables , , and . Hint 1. Which equation should you use for the man’s speed? Because the man’s speed is constant, you may use . ANSWER: Correct Part B What is , the position of the bus as a function of time? Answer symbolically in terms of and . = Extra Credit http://session.masteringphysics.com/myct/assignmentPrintView?displayM… 1 of 57 5/9/2014 8:02 PM Hint 1. Which equation should you use for the bus’s acceleration? Because the bus has constant acceleration, you may use . Recall that . ANSWER: Correct Part C What condition is necessary for the man to catch the bus? Assume he catches it at time . Hint 1. How to approach this problem If the man is to catch the bus, then at some moment in time , the man must arrive at the position of the door of the bus. How would you express this condition mathematically? ANSWER: Correct Part D Inserting the formulas you found for and into the condition , you obtain the following: , or . Intuitively, the man will not catch the bus unless he is running fast enough. In mathematical terms, there is a constraint on the man’s speed so that the equation above gives a solution for that is a real positive number. Find , the minimum value of for which the man will catch the bus. Express the minimum value for the man’s speed in terms of and . = Extra Credit http://session.masteringphysics.com/myct/assignmentPrintView?displayM… 2 of 57 5/9/2014 8:02 PM Hint 1. Consider the discriminant Use the quadratic equation to solve: . What is the discriminant (the part under the radical) of the solution for ? Hint 1. The quadratic formula Recall: If then ANSWER: Hint 2. What is the constraint? To get a real value for , the discriminant must be greater then or equal to zero. This condition yields a constraint that exceed . ANSWER: Correct Part E Assume that the man misses getting aboard when he first meets up with the bus. Does he get a second chance if he continues to run at the constant speed ? = = Extra Credit http://session.masteringphysics.com/myct/assignmentPrintView?displayM… 3 of 57 5/9/2014 8:02 PM Hint 1. What is the general quadratic equation? The general quadratic equation is , where , , and are constants. Depending on the value of the discriminant, , the equation may have two real valued 1. solutions if , 2. one real valued solution if , or 3. two complex valued solutions if . In this case, every real valued solution corresponds to a time at which the man is at the same position as the door of the bus. ANSWER: Correct Adding and Subtracting Vectors Conceptual Question Six vectors (A to F) have the magnitudes and directions indicated in the figure. Part A Which two vectors, when added, will have the largest (positive) x component? Hint 1. Largest x component The two vectors with the largest x components will, when combined, give the resultant with the largest x component. Keep in mind that positive x components are larger than negative x components. No; there is no chance he is going to get aboard. Yes; he will get a second chance Extra Credit http://session.masteringphysics.com/myct/assignmentPrintView?displayM… 4 of 57 5/9/2014 8:02 PM ANSWER: Correct Part B Which two vectors, when added, will have the largest (positive) y component? Hint 1. Largest y component The two vectors with the largest y components will, when combined, give the resultant with the largest y component. Keep in mind that positive y components are larger than negative y components. ANSWER: Correct Part C Which two vectors, when subtracted (i.e., when one vector is subtracted from the other), will have the largest magnitude? Hint 1. Subtracting vectors To subtract two vectors, add a vector with the same magnitude but opposite direction of one of the vectors to the other vector. ANSWER: C and E E and F A and F C and D B and D C and D A and F E and F A and B E and D Extra Credit http://session.masteringphysics.com/myct/assignmentPrintView?displayM… 5 of 57 5/9/2014 8:02 PM Correct Tactics Box 3.1 Determining the Components of a Vector Learning Goal: To practice Tactics Box 3.1 Determining the Components of a Vector. When a vector is decomposed into component vectors and parallel to the coordinate axes, we can describe each component vector with a single number (a scalar) called the component. This tactics box describes how to determine the x component and y component of vector , denoted and . TACTICS BOX 3.1 Determining the components of a vector The absolute value of the x component is the magnitude of the 1. component vector . The sign of is positive if points in the positive x direction; it is negative if points in the negative x direction. 2. 3. The y component is determined similarly. Part A What is the magnitude of the component vector shown in the figure? Express your answer in meters to one significant figure. ANSWER: A and F A and E D and B C and D E and F Extra Credit http://session.masteringphysics.com/myct/assignmentPrintView?displayM… 6 of 57 5/9/2014 8:02 PM Correct Part B What is the sign of the y component of vector shown in the figure? ANSWER: Correct Part C Now, combine the information given in the tactics box above to find the x and y components, and , of vector shown in the figure. Express your answers, separated by a comma, in meters to one significant figure. = 5 positive negative Extra Credit http://session.masteringphysics.com/myct/assignmentPrintView?displayM… 7 of 57 5/9/2014 8:02 PM ANSWER: Correct Conceptual Problem about Projectile Motion Learning Goal: To understand projectile motion by considering horizontal constant velocity motion and vertical constant acceleration motion independently. Projectile motion refers to the motion of unpowered objects (called projectiles) such as balls or stones moving near the surface of the earth under the influence of the earth’s gravity alone. In this analysis we assume that air resistance can be neglected. An object undergoing projectile motion near the surface of the earth obeys the following rules: An object undergoing projectile motion travels horizontally at a constant rate. That is, the x component of its velocity, , is constant. 1. An object undergoing projectile motion moves vertically with a constant downward acceleration whose magnitude, denoted by , is equal to 9.80 near the surface of the earth. Hence, the y component of its velocity, , changes continuously. 2. An object undergoing projectile motion will undergo the horizontal and vertical motions described above from the instant it is launched until the instant it strikes the ground again. Even though the horizontal and vertical motions can be treated independently, they are related by the fact that they occur for exactly the same amount of time, namely the time the projectile is in the air. 3. The figure shows the trajectory (i.e., the path) of a ball undergoing projectile motion over level ground. The time corresponds to the moment just after the ball is launched from position and . Its launch velocity, also called the initial velocity, is . Two other points along the trajectory are indicated in the figure. One is the moment the ball reaches the peak of its trajectory, at time with velocity . Its position at this moment is denoted by or since it is at its maximum height. The other point, at time with velocity , corresponds to the moment just before the ball strikes the ground on the way back down. At this time its position is , also known as ( since it is at its maximum horizontal range. Projectile motion is symmetric about the peak, provided the object lands at the same vertical height from which is was launched, as is the case here. Hence . Part A , = -2,-5 , Extra Credit http://session.masteringphysics.com/myct/assignmentPrintView?displayM… 8 of 57 5/9/2014 8:02 PM How do the speeds , , and (at times ,

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## PHY-102: Motion Exercises Complete the following exercises. 1. Jane is collecting data for a ball rolling down a hill. She measures out a set of different distances and then proceeds to use a stopwatch to find the time it takes the ball to roll each distance. a. What is the independent variable in her experiment? The ball. b. What is the dependent variable in her experiment? The various distances. c. Give one control variable for her experiment. Time. 2. Consider an experiment where you drop an object. a. Briefly describe your proposed experiment. (Make sure it is controlled). b. What would be the independent variable for your experiment? c. What would be the dependent variable for your experiment? d. Give one control variable for your experiment. 3. Consider a freely falling object. a. What is the acceleration (in m/s2) after 5 seconds of fall? b. What is the acceleration (in m/s2) after 10 seconds of fall? c. What is the velocity (in m/s) after 5 seconds of fall? d. What is the velocity (in m/s) of 10 seconds of fall? 4. A sign is hung between two cables as illustrated below. If the sign weighs 350 N, what is the tension (in N) in each cable? 5. A construction worker on a high-rise building is on a platform suspended between two cables as illustrated below. The construction worker weighs 850 N, the plank weighs 450 N, and the tension in the left cable is 550 N. a. What is the tension (in N) in the right cable? b. Explain your answer. 6. Two forces of 50 N and 30 N, respectively, are acting on an object. Find the net force (in N) on the object if … a. the forces are acting in the same direction b. the forces are acting in opposite directions. 7. A box is pulled straight across the floor at a constant speed. It is pulled with a horizontal force of 48 N. a. Find the net force (in N) on the box. b. Find the force of friction (in N) from the floor on the box. c. The person pulling on the box stops pulling and the box comes to a rest. Find the force of friction (in N) on the box when at rest. 8. A bowling ball rolls 32 meters in 0.8 seconds. Find the average speed (in m/s) of the bowling ball in m/s. 9. A car accelerates from 3.5 m/s to 17 m/s in 4.5 seconds. Find the acceleration of the car in m/s2. 10. Rank the following from lowest to highest: a. The support force on you standing in an elevator at rest. b. The support force on you standing in an elevator accelerating upward. c. The support force on you standing in an elevator accelerating downward. 11. Describe the speed and acceleration of the ball released from the top of the track shown in the figure below. 12. Describe the speed and acceleration of the ball released from the top of the track shown in the figure below. 13. Describe the speed and acceleration of the ball released from the top of the track shown in the figure below. 14. You throw a ball upward with a speed of 14 m/s. What is the acceleration of the ball after it leaves your hand? Ignore air resistance and provide an explanation for your answer. 15. How would your answer to the previous question change if you take into account the effects of air resistance? 16. Describe the speed and acceleration of a person sky diving. Include in your explanation a description of the motion before the parachute is opened as well as a description of the motion after the parachute is opened. 17. A net force of 24 N is acting on a 4.0-kg object. Find the acceleration in m/s2. 18. A person pulls horizontally with a force of 64 N on a 14-kg box. There is a force of friction between the box and the floor of 36 N. Find the acceleration of the box in m/s2. Show your work. The remaining questions are multiple-choice questions: 19. One difference between a hypothesis and a theory is that a hypothesis A. is a guess that has not been well tested, whereas a theory is a synthesis of well-tested guesses. B. is testable, whereas a theory is not testable. C. can be revised, whereas a theory cannot be revised. D. is not testable, whereas a theory is testable. 20. A car starts from rest and reached a speed of 24 m/s in 6 seconds. What is the acceleration of the car? A. 144 m B. 6 m/s2 C. 4 m/s2 D. 10 m/s2 E. 0 m/s2 21. Which of the following forces is NOT a contact force? A. Friction force B. Support force C. Force of gravity D. Tension force 22. If you pull horizontally on a desk with a force of 150 N and the desk doesn’t move, the friction force must be 150 N. Now if you pull with 250 N so the desk slides at constant velocity, the friction force is A. more than 150 N, but less than 250 N. B. 250 N. C. more than 250. 23. Suppose a particle is accelerated through space by a constant 10 N force. Suddenly the particle encounters a second force of 10 N in a direction opposite to that of the first force. The particle A. is brought to a rapid halt. B. theoretically accelerates to speeds approaching the speed of light. C. continues at the speed it had when it encountered the second force. D. gradually slows down to a halt. 24. Newton’s First Law of Motion applies to A. objects at rest only. B. moving objects only. C. both moving and nonmoving objects. 25. A freely falling object starts from rest. After falling for 2 seconds, it will have a speed of about A. 5 m/s B. 10 m/s C. 20 m/s D. 40 m/s 26. Suppose an object is in free fall. Each second the object falls A. the same distance as in the second before. B. a larger distance than in the second before. C. with the same instantaneous speed. D. with the same average speed. © 2014. Grand Canyon University. All Rights Reserved.

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## The first sign of physiological sexual response occurs in the vagina within _____ seconds of the initiation of sexual stimulation. Question 22 options: 5-10 30-45 10-30 45-60

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## Your local art museum has asked you to design a gallery dedicated to works of art from one of the following movements: Renaissance Baroque Romanticism Impressionism Modernism Using your outline from Week Four as reference, select one movement for the design of your gallery. You will use Prezi to design your gallery. This program allows you to design your gallery as if you were guiding a visitor to each work of art. You may draw from resources in the textbook, the CourseMate Bonus Images, or digital image resources found in Week One that incorporate the characteristics significant of your chosen movement and time period. You may also choose to reflect back on your weekly art journal entries for any works of art relevant to this movement. Registering to use Prezi is free, and it is highly recommended that you sign up as soon as possible and review the “Learning Prezi is easy” page and the various ways you can design your journal. You can review the Prezi’s “Manual/FAQ” page for detailed information on using the application. In your Prezi, include the following: A title page/slide which includes the name of your gallery, your name, the course, your instructor’s name, and the date submitted. A brief introduction to your gallery, which includes a description of the movement and the time period to which your gallery is dedicated. Six images of works of art that incorporate the characteristics significant to movement and time period. Along with each image of a work of art, include the citation for the work of art. A summary of how the media (materials), methods, and subject are significant to that time period and region, using appropriate art terminology. A summary of how iconographic, historical, political, philosophical, religious, and social factors of the movement are reflected in the work of art. A references page/slide. You must include at least three scholarly sources in addition to the image references. The ProQuest and Credo Reference databases in the Ashford Online Library are helpful sources of information, as are the museum resources provided in Week One. To locate ProQuest and Credo Reference, visit the Ashford Online Library through the tab on the left navigation toolbar and select "Databases by Subject" and then "Visual and Performing Arts." Cite your sources according to APA style as outlined in the Ashford Writing Center. For information regarding APA samples and tutorials, visit the Ashford Writing Center, located within the Learning Resources tab on the left navigation toolbar. Please follow the basic modified APA style citation format in the APA Artwork Citations document to reference works of art. Submit a Word document with your name and the URL to your art journal. Test the URL to ensure that it is accurate and working prior to submitting the Word document. Carefully review the Grading Rubric for the criteria that will be used to evaluate your assignment.

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## Course: PHYS 5426 — Quantum Statistical Physics Assignment #1 Instructor: Gennady Y. Chitov Date Assigned: January 15, 2014 Due Date: January 29, 2014 Problem 1. Prove [a; f(a†)] = @f(a†) @a† (1) [a†; f(a)] = −@f(a) @a (2) for arbitrary function f of operator which admits a series expansion. The Bose creation/ annihilation operators satisfy the standard commutation relations [a; a†] ≡ aa† − a†a = 1 (3) Hint: From Eqs.(1,2) one can figure out the corresponding commutation relations for the powers of creation/annihilation operators and then prove them by the method of mathematical induction. Note that for an arbitrary operator Aˆ: @A^n @A^ = nAˆn−1. Problem 2. In the presence of a constant external force acting on a one-dimensional oscillating particle its Hamiltonian becomes that of the so-called displaced oscillator, and the Schr¨odinger equation ˆH (q) = E (q) of the problem (cf. lecture notes) can be written in terms of dimensionless variables as ( − 1 2 d2 d2 + 1 2 2 − √ 2 ) () = ” () ; (4) where q = √ ~ m! and E = ~!”. a). Write the Schr¨odinger equation (4) in terms of the creation/annihilation operators of the harmonic oscillator ( = 0) = √1 2 (a + a†) (5) d d = √1 2 (a − a†) (6) 1 Via a linear transformation to the new creation/annihilation operators ˜a†; ˜a preserving the bosonic commutation relations for ˜a†; ˜a map the problem (4) of the displaced oscillator onto that of a simple harmonic oscillator with new operators (˜a†; ˜a). b). Find the spectrum (eigenvalues) ” (E) of the displaced oscillator. c). Write the normalized eigenstates |n⟩ of the displaced Hamiltonian (4) via a† and the vacuum state |Θ◦⟩ of the new operators, i.e. ˜a|Θ◦⟩ = 0 (7) d). As follows from the completeness of the oscillator’s eigenstates, the vacuum state of the displaced oscillator |Θ◦⟩ can be related to the simple oscillator’s vacuum |0⟩ (i.e., a|0⟩ = 0) as |Θ◦⟩ = Ω(a†)|0⟩ (8) Find (up to a normalization factor) the operator function Ω(a†) relating two vacua. Hint: in working out Eqs.(7,8), employ Eqs.(1,2). Problem 3. Prove from the standard commutation relations ([ai; a † j ]∓ = ij , etc) that ⟨0|aiaja † ka † l |0⟩ = jkil ± ikjl (9) the sign depending on the statistics. Also calculate the vacuum expectation value ⟨0|ahaiaja † ka † l a† m |0⟩. Problem 4. In the formalism of second quantization the two-particle interaction term of the Hamiltonian for spinless fermions is given by ˆ V = 1 2 ∫ ∫ dxdy ˆ †(x) ˆ †(y)V(x; y) ˆ (y) ˆ (x) (10) For the short-ranged interaction V(x; y) = V(|x−y|) ≡ V(r) = e2 exp(−r)=r find ˆ V in the momentum representation. The field operators and the creation/annihilation operators in the momentum representation are related in the usual way, i.e., ˆ †(x) = ∫ dp (2)3 a†(p)e−ipx (11) Note that the limit → 0 recovers the Coulomb (long-ranged) interaction V(r) = e2=r. What is the Fourier transform V(q) of the Coulomb interaction? 2 Problem 5. The matrix elements of a two-particle interaction from the previous problem can be written as ⟨k3k4|V|k1k2⟩ = (2)3(k1 + k2 − k3 − k4)V(q) (12) where q ≡ k3−k1 is the momentum transfer. Show that the diagonal part of the interaction operator ˆ V found on the previous problem in the k-representation, arises from momentum transfers q = 0 and q = k2−k1. Write down the two interaction terms and identify them as direct (q = 0) and exchange (q = k2 − k1) interactions. Draw the corresponding Feynman diagrams. Problem 6. Find the first correction to the temperature dependence of the chemical potential of the degenerate ideal electron gas, assuming constant particle concentration ⟨N⟩=V . Express the result in terms of T and the zero-temperature chemical potential ◦. For the calculations the following formula (we set kB = 1) can be used: I ≡ ∫ ∞ 0 f(“)d” e(“−)=T + 1 = ∫ 0 f(“)d” + 2 6 T2f′() + O(T4) (13) 3

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## Assignment 1: Coulomb’s Law Due: 8:00am on Wednesday, January 11, 2012 Note: To understand how points are awarded, read your instructor’s Grading Policy. [Switch to Standard Assignment View] Coulomb’s Law Tutorial Learning Goal: To understand how to calculate forces between charged particles, particularly the dependence on the sign of the charges and the distance between them. Coulomb’s law describes the force that two charged particles exert on each other (by Newton’s third law, those two forces must be equal and opposite). The force exerted by particle 2 (with charge ) on particle 1 (with charge ) is proportional to the charge of each particle and inversely proportional to the square of the distance between them: , where and is the unit vector pointing from particle 2 to particle 1. The force vector will be parallel or antiparallel to the direction of , parallel if the product and antiparallel if ; the force is attractive if the charges are of opposite sign and repulsive if the charges are of the same sign. Part A Consider two positively charged particles, one of charge (particle 0) fixed at the origin, and another of charge (particle 1) fixed on the y-axis at . What is the net force on particle 0 due to particle 1? Express your answer (a vector) using any or all of , , , , , , and . ANSWER: = Correct Part B Now add a third, negatively charged, particle, whose charge is (particle 2). Particle 2 fixed on the y-axis at position . What is the new net force on particle 0, from particle 1 and particle 2? Express your answer (a vector) using any or all of , , , , , , , , and . ANSWER: = Correct Part C Particle 0 experiences a repulsion from particle 1 and an attraction toward particle 2. For certain values of and , the repulsion and attraction should balance each other, resulting in no net force. For what ratio is there no net force on particle 0? Express your answer in terms of any or all of the following variables: , , , . ANSWER: = Correct Part D Now add a fourth charged particle, particle 3, with positive charge , fixed in the yz-plane at . What is the net force on particle 0 due solely to this charge? Hint D.1 Find the magnitude of force from particle 3 Hint not displayed Hint D.2 Vector components Hint not displayed Express your answer (a vector) using , , , , , , and . Include only the force caused by particle 3. ANSWER: = Correct Exercise 21.4 You have a pure (24-karat) gold ring with mass . Gold has an atomic mass of and an atomic number of . Part A How many protons are in the ring? ANSWER: = 4.27×1024 Correct Part B What is their total positive charge? ANSWER: = 6.83×105 Correct Part C If the ring carries no net charge, how many electrons are in it? ANSWER: = 4.27×1024 Correct Exercise 21.22 Two point charges are placed on the x-axis as follows: charge = 4.05 is located at 0.197 , and charge = 5.00 is at -0.296 . Part A What is the magnitude of the total force exerted by these two charges on a negative point charge = -6.00 that is placed at the origin? ANSWER: = 2.55×10−6 Correct Part B What is the direction of the total force exerted by these two charges on a negative point charge = -6.00 that is placed at the origin? ANSWER: to the + direction to the – direction perpendicular to the -axis the force is zero Correct Problem 21.66 A charge 4.97 is placed at the origin of an xy-coordinate system, and a charge -1.99 is placed on the positive x-axis at = 3.98 . A third particle, of charge 6.05 is now placed at the point = 3.98 , = 3.01 . Part A Find the x-component of the total force exerted on the third charge by the other two. ANSWER: = 8.66×10−5 Correct Part B Find the y-component of the total force exerted on the third charge by the other two. ANSWER: = −5.40×10−5 Correct Part C Find the magnitude of the total force acting on the third charge. ANSWER: = 1.02×10−4 Correct Part D Find the direction of the total force acting on the third charge. ANSWER: = -0.557 Correct between and +x-axis Problem 21.68 Two identical spheres with mass are hung from silk threads of length , as shown in the figure . Each sphere has the same charge, so . The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges. Part A Suppose that the angle is small, and find the equilibrium separation between the spheres (Hint: If is small, then .) Express your answer in terms of the variables , , and appropriate constants. ANSWER: = Correct

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## HIST 303 Rebels and Renegades Comparative Paper – Conroy & Drakulic In a well-written analysis of about 3 pages, compare and contrast Conroy’s Belfast Diary or Drakulic’s How We Survived Communism and Even Laughed in response to the following question: It can be argued that in the midst of deprivation and hardship, people still exercise considerable agency—or the power to act within one’s particular socio-political context. In fact, living the ordinary can be considered an act of rebellion against an imposing power. That is, people use and experience their lives as resistance to oppression or war. This is sometimes referred to as the “politics of everyday life”. How does this concept of agency play out in these works? In your response, do not simply list examples, but analyze the examples by the authors in relation to the larger themes of the course. A successful assignment will (this is a checklist, so heed it well!!!): * have a solid introduction with an arguable thesis; * be well organized with coherent paragraphs relevant to the thesis; * have a concluding paragraph that concisely and accurately summarizes the paper; * adequately analyze the histories and their connections to each other; * use relevant evidence to substantiate claims; * be analytic, not descriptive; * properly cite and punctuate quotations and evidence; * be paginated; * have an interesting title relevant to the argument (e.g. “Comparative Paper” is unacceptable); * be well written, well edited and well documented. Author Specific Points that discuss everyday activities as resistance Relate to your other Reading (Williams, Hall, Hebdige, etc.) Conroy Drakulic Working Thesis: _____________________________________________________________________ ____________________________________________________________________________________ ****FORMATTING DIRECTIONS: This paper should be 3 – 4 pages (no more), typed, doublespaced, with one-inch margins and 12-point font. This assignment is worth 25% of your grade in this course. You must head your paper with your name and date and include your name and pages (x of x) in a header or footer of each page. At the end of your paper, you must skip four lines then sign with the following: “I attest that the work contained in this document is entirely my own and it numbers x pages.” *****

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