Two Cars on a Curving Road

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1 skiladæmi 4 Due: 11:59pm on Wednesday, September 30, 2015 You will receive no credit for items you complete after the assignment is due. Grading Policy Two Cars on a Curving Road A small car of mass m and a large car of mass 4m drive along a highway at constant speed. They approach a curve of radius R. Both cars maintain the same acceleration a as they travel around the curve. How does the speed of the small car vs compare to the speed of the large car vl as they round the curve? Hint 1. How to approach the problem To solve this problem use proportional reasoning. Find the simplest equation that contains the variables and other known quantities from the problem. Write this equation twice: once for the car of mass m and speed vs and again for the car of mass 4m and speed vl. Write each equation so that all the constants are on one side, cancel any quantities that appear on both sides and keep the variables on the other side. In this problem the variable is v so write your equations in the form v. Finally, compare the two cases presented in the problem. For this question you should find the ratio vs/ vl. Hint 2. Determine which equation to use In this problem you are told that the cars need to experience the same acceleration as they drive around a curve of radius R. You are also given information that relates the masses of the cars. Which equation is best to use to find out information about the speed each car can travel? ac F net Fc v 2 R m mv 2 R a Hint 3. Relating the speeds of the cars You just established the relationship ar v 2. Since a and R are the same for both cars we can conclude that v 2 L v 2 S. Use this equation with what is known about the masses to find the relationship between v 2 S and v 2 L.

2 v 1 S 4 vl v 1 S 2 vl vs vl vs 2vL 4 vs vl ow apply this method for using proportional reasoning to the next problem. ow assume that two identical cars of mass m drive along a highway. One car approaches a curve of radius 2R at speed v. The second car approaches a curve of radius 6R at a speed of 3v. How does the magnitude F1 of the net force exerted on the first car compare to the magnitude F2 of the net force exerted on the second car? Hint 1. Determine the equation to use for the comparison Which of the following equations makes the correct comparison between car 1 and car 2? Hint 1. How to approach the problem To solve this problem use proportional reasoning to find a relation among F, v, and R. Find the simplest equation that contains these variables and other known quantities from the problem. Write this equation twice: once to describe F1, v1, and R1 and again to relate F2, v2, and R2. Write each equation so that all the constants are on one side and the variables are on the other. In this problem the variable is F so write your equations in the form F. Finally, compare the two cases presented in the problem. For this question you should find the ratio F1/ F2. Hint 2. Equation for the net force acting on each car The net force acting on each car as it travels around the highway curves is given by mv Fc 2, R where m is the mass of the car, R is the radius of the curve, and v is the speed of the car. F1v2 1 R1 F1R1 v 2 1 m1v2 1 R1 m1r2 v 2 2 F2v2 2 R2 F2R2 v 2 2 m2v2 2 R2 m2r1 v 2 1

3 F F 2 F F 2 F1 F1 F1 F F2 F2 Being able to use proportional reasoning is a useful skill to have. The more you practice, the better you will become at it. Problem 5.76 Block A in the figure weighs The coefficient of static friction between the block and the surface on which it rests is The weight w is 12.2 and the system is in equilibrium. Find the friction force exerted on block A. f 12.2 Find the maximum weight for which the system will remain in equilibrium. wmax 16.0

4 Problem 5.89 Block A in the figure has a mass of 4.50, and block B has mass 13.0 kg. The coefficient of kinetic friction between block B and the horizontal surface is kg What is the mass of block C if block B is moving to the right and speeding up with an acceleration 1.90? m/s 2 mc 12.2 kg What is the tension in each cord when block B has this acceleration? TAB 52.7 Part C TBC 96.5

5 ± The Work Done in Pulling a Supertanker Two tugboats pull a disabled supertanker. Each tug exerts a constant force of , one at an angle 11.0 west of north, and the other at an angle 11.0 east of north, as they pull the tanker a distance toward the north. km What is the total work done by the two tugboats on the supertanker? Express your answer in joules, to three significant figures. Hint 1. How to approach the problem There are two ways to calculate the total work done on an object when several forces act on it. You can compute the quantities of work done on the object by each force and then add them together. Alternatively, you can compute the work done on the object by the net force acting on it. The hints that follow are meant to help you to calculate the total work using the first method. Hint 2. Find the work done by one tugboat What is the work done on the tanker by the tugboat that exerts a force in the direction west of north? Express your answer in joules, to three significant figures. Hint 1. The definition of work The work W done by a constant force F acting on an object that undergoes a straight line displacement d is given by the formula W Fdcos ϕ where ϕ is the angle between the direction of the force and the direction of displacement., ± All Work and o Play Learning Goal: To be able to calculate work done by a constant force directed at different angles relative to displacement If an object undergoes displacement while being acted upon by a force (or several forces), it is said that work is being done on the object. If the object is moving in a straight line and the displacement and the force are known, the work done by the force can be calculated as

6 W F s F s cos θ W F s θ F where is the work done by force on the object that undergoes displacement directed at angle relative to. cos θ ote that depending on the value of, the work done can be positive, negative, or zero. In this problem, you will practice calculating work done on an object moving in a straight line. The first series of questions is related to the accompanying figure., What can be said about the sign of the work done by the force F 1? It is positive. It is negative. It is zero. There is not enough information to answer the question. When, the cosine of θ is zero, and therefore the work done is zero. θ 90 What can be said about the work done by force F 2? It is positive. It is negative. It is zero. 0 < θ < 90 cos θ When, is positive, and so the work done is positive. Part C

7 The work done by force F 3 is positive negative zero 90 < θ < 180 cos θ When, is negative, and so the work done is negative. Part D The work done by force F 4 is positive negative zero Part E The work done by force F 5 is positive negative zero Part F The work done by force F 6 is positive negative zero

8 Part G The work done by force F 7 is positive negative zero In the next series of questions, you will use the formula W F s F s cos θ to calculate the work done by various forces on an object that moves 160 meters to the right. Part H Find the work W done by the 18 newton force. Use two significant figures in your answer. Express your answer in joules. W 2900 Part I Find the work W done by the 30 newton force. Use two significant figures in your answer. Express your answer in joules. W 4200

9 Part Find the work W done by the 12 newton force. Use two significant figures in your answer. Express your answer in joules. W 1900 Part K Find the work W done by the 15 newton force. Use two significant figures in your answer. Express your answer in joules. W 1800 Exercise 6.42 kg k /m A block of ice of mass 4.50 is placed against a horizontal spring that has force constant 190 and is compressed a distance m. The spring is released and accelerates the block along a horizontal surface. You can ignore friction and the mass of the spring. Calculate the work done on the block by the spring during the motion of the block from its initial position to where the spring has returned to its uncompressed length. Express your answer using two significant figures. W What is the speed of the block after it leaves the spring? Express your answer using two significant figures. v 0.14 m/s

10 Exercise 6.34 cm To stretch a spring 9.00 from its unstretched length, 19.0 of work must be done. What is the force constant of this spring? k 4690 /m If you need to use the value of the spring constant 'k' in subsequent parts, please use the unrounded full precision value and not the one you submitted for this part rounded using three significant figures. What magnitude force is needed to stretch the spring 9.00 cm from its unstretched length? F 422 Part C How much work must be done to compress this spring 4.00 cm from its unstretched length? W 3.75 Part D What force is needed to stretch it this distance? F 188 Score Summary:

11 Your score on this assignment is 101%. You received 7.07 out of a possible total of 7 points.

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