## Assignment 7 Due: 11:59pm on Friday, March 21, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 8.5 The figure shows two balls of equal mass moving in vertical circles. Part A Is the tension in string A greater than, less than, or equal to the tension in string B if the balls travel over the top of the circle with equal speed? ANSWER: Correct The tension in string A is less than the tension in string B. The tension in string A is equal to the tension in string B. The tension in string A is greater than the tension in string B. Part B Is the tension in string A greater than, less than, or equal to the tension in string B if the balls travel over the top of the circle with equal angular velocity? ANSWER: Correct A Mass on a Turntable: Conceptual A small metal cylinder rests on a circular turntable that is rotating at a constant rate, as illustrated in the diagram. Part A Which of the following sets of vectors best describes the velocity, acceleration, and net force acting on the cylinder at the point indicated in the diagram? The tension in string A is less than the tension in string B. The tension in string A is equal to the tension in string B. The tension in string A is greater than the tension in string B. Typesetting math: 100% Hint 1. The direction of acceleration can be determined from Newton’s second law According to Newton’s second law, the acceleration of an object has the same direction as the net force acting on that object. ANSWER: Correct Part B Let be the distance between the cylinder and the center of the turntable. Now assume that the cylinder is moved to a new location from the center of the turntable. Which of the following statements accurately describe the motion of the cylinder at the new location? Check all that apply. a b c d e R R/2 Typesetting math: 100% Hint 1. Find the speed of the cylinder Find the speed of the cylinder at the new location. Assume that the cylinder makes one complete turn in a period of time . Express your answer in terms of and . ANSWER: Hint 2. Find the acceleration of the cylinder Find the magnitude of the acceleration of the cylinder at the new location. Assume that the cylinder makes one complete turn in a period of time . Express your answer in terms of and . Hint 1. Centripetal acceleration Recall that the acceleration of an object that moves in a circular path of radius with constant speed has magnitude given by . Note that both the velocity and radius of the trajectory change when the cylinder is moved. ANSWER: ANSWER: v T R T v = R T a T R T r v a = v2 r a = 22R T 2 Typesetting math: 100% Correct Accelerating along a Racetrack A road race is taking place along the track shown in the figure . All of the cars are moving at constant speeds. The car at point F is traveling along a straight section of the track, whereas all the other cars are moving along curved segments of the track. Part A Let be the velocity of the car at point A. What can you say about the acceleration of the car at that point? Hint 1. Acceleration along a curved path The speed of the cylinder has decreased. The speed of the cylinder has increased. The magnitude of the acceleration of the cylinder has decreased. The magnitude of the acceleration of the cylinder has increased. The speed and the acceleration of the cylinder have not changed. v A Typesetting math: 100% Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object’s acceleration is always perpendicular to the velocity vector at each point along the curved path and is directed toward the center of curvature of the path. ANSWER: Correct Part B Let be the velocity of the car at point C. What can you say about the acceleration of the car at that point? Hint 1. Acceleration along a curved path Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object’s acceleration is always perpendicular to the velocity vector at each point along the curved path and is directed toward the center of curvature of the path. ANSWER: v v The acceleration is parallel to . The acceleration is perpendicular to and directed toward the inside of the track. The acceleration is perpendicular to and directed toward the outside of the track. The acceleration is neither parallel nor perpendicular to . The acceleration is zero. v A v A v A v A v C v v Typesetting math: 100% Correct Part C Let be the velocity of the car at point D. What can you say about the acceleration of the car at that point? Hint 1. Acceleration along a curved path Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object’s acceleration is always perpendicular to the velocity vector at each point along the curved path and is directed toward the center of curvature of the path. ANSWER: Correct The acceleration is parallel to . The acceleration is perpendicular to and pointed toward the inside of the track. The acceleration is perpendicular to and pointed toward the outside of the track. The acceleration is neither parallel nor perpendicular to . The acceleration is zero. v C v C v C v C v D v v The acceleration is parallel to . The acceleration is perpendicular to and pointed toward the inside of the track. The acceleration is perpendicular to and pointed toward the outside of the track. The acceleration is neither parallel nor perpendicular to . The acceleration is zero. v D v D v D v D Typesetting math: 100% Part D Let be the velocity of the car at point F. What can you say about the acceleration of the car at that point? Hint 1. Acceleration along a straight path The velocity of an object that moves along a straight path is always parallel to the direction of the path, and the object has a nonzero acceleration only if the magnitude of its velocity changes in time. ANSWER: Correct Part E Assuming that all cars have equal speeds, which car has the acceleration of the greatest magnitude, and which one has the acceleration of the least magnitude? Use A for the car at point A, B for the car at point B, and so on. Express your answer as the name the car that has the greatest magnitude of acceleration followed by the car with the least magnitude of accelation, and separate your answers with a comma. Hint 1. How to approach the problem Recall that the magnitude of the acceleration of an object that moves at constant speed along a curved path is inversely proportional to the radius of curvature of the path. ANSWER: v F The acceleration is parallel to . The acceleration is perpendicular to and pointed toward the inside of the track. The acceleration is perpendicular to and pointed toward the outside of the track. The acceleration is neither parallel nor perpendicular to . The acceleration is zero. v F v F v F v F Typesetting math: 100% Correct Part F Assume that the car at point A and the one at point E are traveling along circular paths that have the same radius. If the car at point A now moves twice as fast as the car at point E, how is the magnitude of its acceleration related to that of car E. Hint 1. Find the acceleration of the car at point E Let be the radius of the two curves along which the cars at points A and E are traveling. What is the magnitude of the acceleration of the car at point E? Express your answer in terms of the radius of curvature and the speed of car E. Hint 1. Uniform circular motion The magnitude of the acceleration of an object that moves with constant speed along a circular path of radius is given by . ANSWER: Hint 2. Find the acceleration of the car at point A If , what is the acceleration of the car at point A? Let be the radius of the two curves along which the cars at points A and E are traveling. Express your answer in terms of the speed of the car at E and the radius . r aE r vE a v r a = v2 r aE = vE 2 r vA = 2vE aA r vE r Typesetting math: 100% Hint 1. Uniform circular motion The magnitude of the acceleration of an object that moves with constant speed along a circular path of radius is given by . ANSWER: ANSWER: Correct Problem 8.5 A 1300 car takes a 50- -radius unbanked curve at 13 . Part A What is the size of the friction force on the car? Express your answer to two significant figures and include the appropriate units. ANSWER: v r a = v2 r aA = 4vE 2 r The magnitude of the acceleration of the car at point A is twice that of the car at point E. The magnitude of the acceleration of the car at point A is the same as that of the car at point E. The magnitude of the acceleration of the car at point A is half that of the car at point E. The magnitude of the acceleration of the car at point A is four times that of the car at point E. kg m m/s Typesetting math: 100% Correct Problem 8.10 It is proposed that future space stations create an artificial gravity by rotating. Suppose a space station is constructed as a 1600- -diameter cylinder that rotates about its axis. The inside surface is the deck of the space station. Part A What rotation period will provide “normal” gravity? Express your answer with the appropriate units. ANSWER: Correct Problem 8.7 In the Bohr model of the hydrogen atom, an electron orbits a proton at a distance of . The proton pulls on the electron with an electric force of . Part A How many revolutions per second does the electron make? Express your answer with the appropriate units. ANSWER: fs = 4400 N m T = 56.8 s (mass m = 9.1 × 10−31 kg) 5.3 × 10−11 m 8.2 × 10−8 N Typesetting math: 100% Correct Problem 8.14 The weight of passengers on a roller coaster increases by 56 as the car goes through a dip with a 38 radius of curvature. Part A What is the car’s speed at the bottom of the dip? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 8.18 While at the county fair, you decide to ride the Ferris wheel. Having eaten too many candy apples and elephant ears, you find the motion somewhat unpleasant. To take your mind off your stomach, you wonder about the motion of the ride. You estimate the radius of the big wheel to be 14 , and you use your watch to find that each loop around takes 24 . Part A What is your speed? Express your answer to two significant figures and include the appropriate units. ANSWER: 6.56×1015 rev s % m v = 14 ms m s v = 3.7 ms Typesetting math: 100% Correct Part B What is the magnitude of your acceleration? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the ratio of your weight at the top of the ride to your weight while standing on the ground? Express your answer using two significant figures. ANSWER: Correct Part D What is the ratio of your weight at the bottom of the ride to your weight while standing on the ground? Express your answer using two significant figures. ANSWER: a = 0.96 m s2 = 0.90 wtop FG Typesetting math: 100% Correct Enhanced EOC: Problem 8.46 A heavy ball with a weight of 120 is hung from the ceiling of a lecture hall on a 4.4- -long rope. The ball is pulled to one side and released to swing as a pendulum, reaching a speed of 5.6 as it passes through the lowest point. You may want to review ( pages 201 – 204) . For help with math skills, you may want to review: Solutions of Systems of Equations Part A What is the tension in the rope at that point? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Start by drawing a free-body diagram indicating the forces acting on the ball when it is at its lowest point. Choose a coordinate system. What is the direction of the acceleration in your chosen coordinate system? What is the magnitude of the acceleration for the mass, which is moving in a circular path? What is Newton’s second law applied to the mass at the bottom of its swing? Make sure to use your coordinate system when determining the signs of all the forces and the acceleration. What is the tension in the rope at this point? ANSWER: = 1.1 wbottom FG N m m/s T = 210 N Typesetting math: 100% Correct Problem 8.43 In an amusement park ride called The Roundup, passengers stand inside a 16.0 -diameter rotating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane, as shown in the figure . Part A Suppose the ring rotates once every 4.80 . If a rider’s mass is 54.0 , with how much force does the ring push on her at the top of the ride? Express your answer with the appropriate units. ANSWER: Correct Part B m s kg 211 N Typesetting math: 100% Suppose the ring rotates once every 4.80 . If a rider’s mass is 54.0 , with how much force does the ring push on her at the bottom of the ride? Express your answer with the appropriate units. ANSWER: Correct Part C What is the longest rotation period of the wheel that will prevent the riders from falling off at the top? Express your answer with the appropriate units. ANSWER: Correct Conceptual Question 9.9 A 2 object is moving to the right with a speed of 1 when it experiences an impulse of 6 . Part A What is the object’s speed after the impulse? Express your answer as an integer and include the appropriate units. ANSWER: s kg 1270 N 5.68 s kg m/s i ^ Ns i ^ v = 4 ms Typesetting math: 100% Correct Part B What is the object’s direction after the impulse? ANSWER: Correct Conceptual Question 9.10 A 2 object is moving to the right with a speed of 2 when it experiences an impulse of -6 . Part A What is the object’s speed after the impulse? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part B What is the object’s direction after the impulse? to the right to the left kg m/s i ^ Ns i ^ v = 1 ms Typesetting math: 100% ANSWER: Correct Problem 9.5 Part A In the figure , what value of gives an impulse of 6.4 ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct to the right to the left Fmax Ns Fmax = 1.6×103 N Typesetting math: 100% Impulse on a Baseball Learning Goal: To understand the relationship between force, impulse, and momentum. The effect of a net force acting on an object is related both to the force and to the total time the force acts on the object. The physical quantity impulse is a measure of both these effects. For a constant net force, the impulse is given by . The impulse is a vector pointing in the same direction as the force vector. The units of are or . Recall that when a net force acts on an object, the object will accelerate, causing a change in its velocity. Hence the object’s momentum ( ) will also change. The impulse-momentum theorem describes the effect that an impulse has on an object’s motion: . So the change in momentum of an object equals the net impulse, that is, the net force multiplied by the time over which the force acts. A given change in momentum can result from a large force over a short time or a smaller force over a longer time. In Parts A, B, C consider the following situation. In a baseball game the batter swings and gets a good solid hit. His swing applies a force of 12,000 to the ball for a time of . Part A Assuming that this force is constant, what is the magnitude of the impulse on the ball? Enter your answer numerically in newton seconds using two significant figures. ANSWER: Correct We often visualize the impulse by drawing a graph of force versus time. For a constant net force such as that used in the previous part, the graph will look like the one shown in the figure. (F J J = F) t J N * s kg * m/s p = mv )p = J = F) t N 0.70 × 10−3 s J J = 8.4 N * s Typesetting math: 100% Part B The net force versus time graph has a rectangular shape. Often in physics geometric properties of graphs have physical meaning. ANSWER: Correct The assumption of a constant net force is idealized to make the problem easier to solve. A real force, especially in a case like the one presented in Parts A and B, where a large force is applied for a short time, is not likely to be constant. A more realistic graph of the force that the swinging bat applies to the baseball will show the force building up to a maximum value as the bat comes into full contact with the ball. Then as the ball loses contact with the bat, the graph will show the force decaying to zero. It will look like the graph in the figure. For this graph, the length height area slope of the rectangle corresponds to the impulse. Typesetting math: 100% Part C If both the graph representing the constant net force and the graph representing the variable net force represent the same impulse acting on the baseball, which geometric properties must the two graphs have in common? ANSWER: maximum force area slope Typesetting math: 100% Correct When the net force varies over time, as in the case of the real net force acting on the baseball, you can simplify the problem by finding the average net force acting on the baseball during time . This average net force is treated as a constant force that acts on the ball for time . The impulse on the ball can then be found as . Graphically, this method states that the impulse of the baseball can be represented by either the area under the net force versus time curve or the area under the average net force versus time curve. These areas are represented in the figure as the areas shaded in red and blue respectively. The impulse of an object is also related to its change in momentum. Once the impulse is known, it can be used to find the change in momentum, or if either the initial or final momentum is known, the other momentum can be found. Keep in mind that . Because both impulse and momentum are vectors, it is essential to account for the direction of each vector, even in a one-dimensional problem. Part D Assume that a pitcher throws a baseball so that it travels in a straight line parallel to the ground. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. Define the direction the pitcher originally throws the ball as the +x direction. ANSWER: F avg )t )t J = F )t avg J = )p = m(vf − vi ) Typesetting math: 100% Correct Part E Now assume that the pitcher in Part D throws a 0.145- baseball parallel to the ground with a speed of 32 in the +x direction. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. What is the ball’s velocity just after leaving the bat if the bat applies an impulse of to the baseball? Enter your answer numerically in meters per second using two significant figures. ANSWER: Correct The negative sign in the answer indicates that after the bat hits the ball, the ball travels in the opposite direction to that defined to be positive. Problem 9.9 A 2.6 object is moving to the right with a speed of 1.0 when it experiences the force shown in the figure. The impulse on the ball caused by the bat will be in the positive negative x direction. kg m/s −8.4 N * s v = -26 m/s kg m/s Typesetting math: 100% Part A What is the object’s speed after the force ends? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the object’s direction after the force ends? ANSWER: Correct Enhanced EOC: Problem 9.27 A tennis player swings her 1000 g racket with a speed of 11.0 . She hits a 60 g tennis ball that was approaching her at a speed of 19.0 . The ball rebounds at 41.0 . You may want to review ( pages 226 – 232) . For help with math skills, you may want to review: v = 0.62 ms to the right to the left m/s m/s m/s Typesetting math: 100% Solving Algebraic Equations Part A How fast is her racket moving immediately after the impact? You can ignore the interaction of the racket with her hand for the brief duration of the collision. Express your answer with the appropriate units. Hint 1. How to approach the problem Given that you can ignore the interaction of the racket with her hand during the collision, what is conserved during the collision? Draw a picture indicating the direction of the racket and ball before the collision and a separate picture for after the collision. Place a coordinate system on your pictures, indicating the positive x direction. Keeping in mind that velocity can be either positive or negative in your coordinate system, what is the initial momentum of the ball–racket system? What is the final momentum of the ball–racket system in terms of the velocity of the racket after the collision? Using conservation of momentum, what are the velocity and speed of the racket after the collision? ANSWER: Correct Part B If the tennis ball and racket are in contact for 8.00 , what is the average force that the racket exerts on the ball? Express your answer with the appropriate units. Hint 1. How to approach the problem How is the impulse on the ball related to the change in momentum of the ball? What is the change in momentum of the ball? How are the impulse on the ball and the collision time related to the average force on the ball? 7.40 ms ms Typesetting math: 100% ANSWER: Correct Problem 9.14 A 2.00×104 railroad car is rolling at 6.00 when a 6000 load of gravel is suddenly dropped in. Part A What is the car’s speed just after the gravel is loaded? Express your answer with the appropriate units. ANSWER: Correct Problem 9.17 A 330 bird flying along at 5.0 sees a 9.0 insect heading straight toward it with a speed of 34 (as measured by an observer on the ground, not by the bird). The bird opens its mouth wide and enjoys a nice lunch. Part A What is the bird’s speed immediately after swallowing? Express your answer to two significant figures and include the appropriate units. ANSWER: 450 N kg m/s kg 4.62 ms g m/s g m/s Typesetting math: 100% Correct Problem 9.20 A 50.0 archer, standing on frictionless ice, shoots a 200 arrow at a speed of 200 . Part A What is the recoil speed of the archer? Express your answer with the appropriate units. ANSWER: Correct Problem 9.25 A 40.0 ball of clay traveling east at 4.50 collides and sticks together with a 50.0 ball of clay traveling north at 4.50 . Part A What is the speed of the resulting ball of clay? Express your answer with the appropriate units. ANSWER: v = 4.0 ms kg g m/s 0.800 ms g m/s g m/s 3.20 ms Typesetting math: 100% Correct Problem 9.32 A particle of mass is at rest at . Its momentum for is given by , where is in . Part A Find an expression for , the force exerted on the particle as a function of time. Express your answer in terms of the given quantities. ANSWER: Correct Problem 9.37 Most geologists believe that the dinosaurs became extinct 65 million years ago when a large comet or asteroid struck the earth, throwing up so much dust that the sun was blocked out for a period of many months. Suppose an asteroid with a diameter of 2.0 and a mass of 1.2×1013 hits the earth with an impact speed of 4.5×104 . Part A What is the earth’s recoil speed after such a collision? (Use a reference frame in which the earth was initially at rest.) Assume that . Express your answer to two significant figures and include the appropriate units. ANSWER: m t = 0 t > 0 px = 6t2 kgm/s t s Fx(t) Fx = 12t N km kg m/s MEarth= 5.98 × 1024 kg = 9.0×10−8 v ms Typesetting math: 100% Correct Part B What percentage is this of the earth’s speed around the sun? (Use the astronomical data in the textbook.) Express your answer using two significant figures. ANSWER: Correct Problem 9.42 One billiard ball is shot east at 1.8 . A second, identical billiard ball is shot west at 1.2 . The balls have a glancing collision, not a head-on collision, deflecting the second ball by 90 and sending it north at 1.50 . Part A What is the speed of the first ball after the collision? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the direction of the first ball after the collision? Give the direction as an angle south of east. = 3.0×10−10 of v % the earth’s speed m/s m/s 1 m/s v = 1.6 ms Typesetting math: 100% Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 9.49 Two 490 blocks of wood are 2.0 apart on a frictionless table. A 12 bullet is fired at 420 toward the blocks. It passes all the way through the first block, then embeds itself in the second block. The speed of the first block immediately afterward is 5.6 . Part A What is the speed of the second block after the bullet stops? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 99.5%. You received 156.21 out of a possible total of 157 points. = 68 1 g m g m/s m/s v = 4.6 ms Typesetting math: 100%

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