3. Springs; Elastic Forces & Energy.notebook

Transcription

1 1. When a mass is placed on a spring with a spring constant of 60.0 newtons per meter, the spring is compressed meter. How much energy is stored in the spring? J J J J 2. A spring gains 2.34 joules of elastic potential energy as it is compressed meter from its equilibrium position. What is the spring constant, k, of this spring? N/m N/m N/m N/m 3. A spring with a spring constant of 4.0 newtons per meter is compressed by a force of 1.2 newtons. What is the total elastic potential energy stored in this compressed spring? J J J J 4. A child does 0.20 joule of work to compress the spring in a pop up toy. If the mass of the toy is kilogram, what is the maximum vertical height that the toy can reach after the spring is released? m m m m 1

2 5. The diagram below shows a toy cart possessing 16 joules of kinetic energy traveling on a frictionless, horizontal surface toward a horizontal spring. If the cart comes to rest after compressing the spring a distance of 1.0 meter, what is the spring constant of the spring? N/m N/m N/m N/m The kinetic energy of the cart is transferred into the potential energy of the spring by compression. Given the kinetic energy of the cart and assuming no loss of energy, the spring must gain the 16 Joules of energy transferred by the cart. We can then apply the potential energy of a spring equation and solve for the spring constant. 2

3 6. An unstretched spring has a length of 10. centimeters. When the spring is stretched by a force of 16 newtons, its length is increased to 18 centimeters. What is the spring constant of this spring? N/cm N/cm N/cm N/cm 7. A spring in a toy car is compressed a distance, x. When released, the spring returns to its original length, transferring its energy to the car. Consequently, the car having mass m moves with speed v. Derive the spring constant, k, of the car s spring in terms of m, x, and v. [Assume an ideal mechanical system with no loss of energy.]

4 8. Which graph best represents the relationship between the elastic potential energy stored in a spring and its elongation from equilibrium? 9. The diagram below represents a spring hanging vertically that stretches meter when a 5.0 newton block is attached. The spring block system is at rest in the position shown. The value of the spring constant is N/m N/m N/m N/m The 5.0 N force on the spring changes its displacement length by m. (7.5 cm) 4

5 10. A spring with a spring constant of 80. newtons per meter is displaced 0.30 meter from its equilibrium position. The potential energy stored in the spring is J J J J 11. The graph below represents the relationship between the force applied to a spring and spring elongation for four different springs. Which spring has the greatest spring constant? 1. spring A 2. spring B 3. spring C 4. spring D The slope of each of the lines in the graph represents each spring s constant. Line A with the greatest slope has the greatest spring constant of the choices listed. F=kx 5

6 12. When a mass is placed on a spring with a spring constant of 15 newtons per meter, the spring is compressed 0.25 meter. How much elastic potential energy is stored in the spring? J J J J 13. A vertical spring meter long is elongated to a length of meter when a 1.00 kilogram mass is attached to the bottom of the spring. The spring constant of this spring is N/m N/m N/m N/m 14. The spring in a scale in the produce department of a supermarket stretches meter when a watermelon weighing newtons is placed on the scale. The spring constant for this spring is N/m N/m N/m N/m When a force is applied to a spring, that spring will elongate or stretch a certain amount. The amount of elongation is a function of the spring s constant and can be found by applying Hooke s law. We can also transpose Hooke s equation to find the spring constant. We must first convert the mass attached to the spring to the force this mass produces which is equivalent to its weight. 6

7 15. A 1.0 kilogram mass is suspended from each spring. If each mass is at rest, how does the potential energy stored in spring A compare to the potential energy stored in spring B? 1. The potential energy stored in spring A is greater than the potential energy stored in spring B. 2. The potential energy stored in spring A is less than the potential energy stored in spring B. 3. The potential energy stored in spring A is the same as the potential energy stored in spring B. 16. k = F /x Pull to stretch ratio Base your answer to question on the information and graph above. The graph represents the relationship between the force applied to each of two springs, A and B, and their elongations. What physical quantity is represented by the slope of each line? 1. the spring constant 2. the potential energy of the spring 3. the period of the spring 4. the maximum elongation of the spring Use PE = 1/2kx 2 where x represents the elongation or displacement (deformation) of the spring, and k = the spring constant. From the graph, we can see that spring B has a greater elongation and, therefore, the square of that elongation produces more potential energy than spring A. k = F /x Pull to stretch ratio 7

8 17. As shown in the diagram, a 0.50 meter long spring is stretched from its equilibrium position to a length of 1.00 meter by a weight. If 15 joules of energy are stored in the stretched spring, what is the value of the spring constant? N/m N/m N/m N/m x =.5 m The potential energy stored in a spring is a function of the spring constant and the amount of elongation or compression of the spring. Apply the equation for calculating the potential energy of a spring. 8

9 18. A 5 newton force causes a spring to stretch 0.2 meter. What is the potential energy stored in the stretched spring? 1. 1 J J J J 19. A 10. newton force is required to hold a stretched spring 0.20 meter from its rest position. What is the potential energy stored in the stretched spring? J J J J PE = 1/2 kx 2 =.5(25)(.2) 2 =.5J The spring constant (k) is related to the force used and the change in the spring length from the equilibrium position. (Each spring can have a different spring constant.) 9

10 20. The graph below shows elongation as a function of the applied force for two springs, A and B. Compared to the spring constant for spring A, the spring constant for spring Bis 1. smaller 2. larger 3. the same The graph is comparing two variables of Hooke s Law, but look at which variable corresponds to which axis! When a force is applied to a spring, the elongation of the spring is related both to the applied force and to the spring constant of the spring. The spring constant is a measure of the "springiness" of the spring. As the spring constant k increases, a greater force must be applied to produce the same elongation as a spring with a smaller spring constant. Look at the graph. The application of a given force to spring B produces a smaller elongation than it does for spring A. That means spring B must be stiffer than spring A; that is, the spring constant for B must be greater than the spring constant for spring A. 21. As a spring is stretched, its elastic potential energy 1. decreases 2. increases (PE = 1/2kx 2 ) 3. remains the same Since the elastic potential energy of a spring is directly proportional to the square of its deformation or displacement, the spring s elastic potential energy increases 10

11 22. Which graph best represents the elastic potential energy stored in a spring (PE s ) as a function of its elongation, x? 23. The graph below represents the relationship between the force applied to a spring and the compression (displacement) of the spring. What is the spring constant for this spring? N/m N/m N/m N/m The spring constant is the ratio of the force applied to a spring and the change in its length. An applied force will compress or stretch a spring. There is a direct linear relationship between the magnitude of the applied force and the change in the spring's length: F = kx, so long as the elastic limit of the spring is not exceeded. The constant of proportionality, k, called the spring constant, is the slope of the force versus compression graph. Choose two points on the graph, say (0 m, 0 N) and (0.40 m, 1.00 N), and use them to calculate the slope, k: 11

12 24. In the diagram below, an average force of 20. newtons is used to pull back the string of a bow 0.60 meter. As the arrow leaves the bow, its kinetic energy is J J J J The work done on displacing the arrow builds up its potential energy. Look for an equation relating work, force, and displacement. Reason The stored energy becomes the kinetic energy. Pulling back on the bow string stores displaces the arrow and builds up the potential energy in the system. This potential energy is converted to kinetic energy when the arrow is released. The magnitude of the force applied to the arrow is F = 20. N, and this force displaces the arrow by d = 0.60 m. Apply the work equation W = Fd to calculate the amount of work accomplished: W = Fd = (20. N)(0.60 m) = 12 N m = 12 J Thus, when the arrow is released, its 12 joules of potential energy is converted to 12 joules of kinetic energy. 25. The unstretched spring in the diagram has a length of 0.40 meter and a spring constant k. A weight is hung from the spring, causing it to stretch to a length of 0.60 meter. How many joules of elastic potential energy are stored in this stretched spring? k k k k x =.2 m Apply the equation for the potential energy stored in a spring. An elastic spring that is compressed or stretched beyond its equilibrium position stores potential energy. The elongation of the given spring is x = 0.60 m 0.40 m = 0.20 m. Apply the equation that expresses the amount of elastic potential energy stored in the spring in terms of the spring constant, k: 12

13 26. In the diagram below, an ideal pendulum released from position A swings freely to position B. As the pendulum swings from A to B, its total mechanical energy 1. decreases, then increases 2. increases, only 3. increases, then decreases 4. remains the same Disregarding friction, the total mechanical energy of the pendulum remains the same as the pendulum bob moves from point A to point B and back. The energy transforms from kinetic to potential, but there is no loss or gain in mechanical energy. 27. A pendulum is made from a 7.50 kilogram mass attached to a rope connected to the ceiling of a gymnasium. The mass is pushed to the side until it is at position A, 1.5 meters higher than its equilibrium position. After it is released from rest at position A, the pendulum moves freely back and forth between positions A and B, as shown in the diagram. What is the total amount of kinetic energy that the mass has as it swings freely through its equilibrium position? [Neglect friction.] J J J J Apply the principle of conservation of energy. Applying the concept of conservation of energy: the potential energy that the pendulum mass has at the highest position equals the kinetic energy that it has moving through the equilibrium position. 13

14 28. A pendulum is pulled to the side and released from rest. Which graph best represents the relationship between the gravitational potential energy of the pendulum and its displacement from its point of release? Think of the pendulum s swing as the pendulum bob increases and decreases in height The graph in choice (4) indicates the correct potential energy as the displacement of the pendulum changes throughout its swing. The potential energy is highest when the displacement is highest; potential energy is lowest when the displacement is lowest. 29. In the diagram, an ideal pendulum released from point A swings freely through point B. Compared to the pendulum's kinetic energy at A, its potential energy at B is 1. half as great 2. twice as great 3. the same 4. four times as great The pendulum has only potential energy at point A, and only kinetic energy at point B. Apply the principle of conservation of energy. At the highest point in its swing, point A, the pendulum is motionless and all of the energy it possess is in the form of potential energy. When the pendulum reaches the lowest point in its swing, point B, all of this potential energy is converted to kinetic energy. 14