Harmonic Motion: Exercises 1. The following is a list of forces, each of which is the net external force acting on an object with mass number m that is free to move in onedimension only. Assume that s is the position relative to the origin, s 0 = 0, at which the object is in equilibrium. The symbols a, b, c, and d are positive constants, and F o is a small constant force directed in the positive direction. Identify every net force in the list that would cause the object on which it is acting to undergo simple harmonic motion if the object were displaced from the origin and released. (a) F net = as (b) F net = ds (c) F net = (a + b)s (d) F net = (a + b + c)s (e) F net = as + F o (f) F net = (be c/d )s (g) F net = [(a + b)/(d + c)]s (h) F net = as/ s (i) None of the above. The following information pertains to questions 2-10. An object with mass number m, free to move one-dimensionally on a horizontal, frictionless surface, is subjected to a restoring force of magnitude ks where s is the object s position relative to its equilibrium position, defined as s 0 = 0. The direction of this restoring force is such that it always pushes or pulls the object toward its equilibrium position. The object is displaced and released at rest from position s i and undergoes simple harmonic motion with period T. The total energy associated with the motion of the object is E. As the object passes through its equilibrium position, its velocity has magnitude v 0. 2. If the spring constant, k, had been slightly larger, then, compared to v 0, the magnitude of the object s velocity as it passes through its equilibrium position would have been 1
3. If the spring constant, k, had been slightly smaller, then, compared with T, the period of the motion would have been 4. If the spring constant, k, had been slightly larger, then, compared with E, the total energy associated with the motion of the object would have been 5. If the object had been released (also at rest) from position s < s i, then, compared with v 0, the magnitude of the object s velocity as it passed through its equilibrium position would have been 6. If the object had been released (also at rest) from position s > s i, then, compared with T, the period of the motion would have been 7. If the object had been released (also at rest) from position s < s i, then, compared with E, the total energy of the motion would have been 2
8. If the mass of the object had been slightly less than m, then compared with v 0, the magnitude of the object s velocity as it passed through its equilibrium position would have been 9. If the mass of the object had been slightly less than m, then compared with T, the period of the motion would have been 10. If the mass of the object had been slightly less than m, then compared with E, the total energy of the motion would have been The following information pertains to questions 11-22. A simple pendulum is made by tying an object with mass number m to the end of a string of length l, the other end of which is tied to the ceiling. The object is pulled aside until the string makes a small angle θ with the vertical, held at rest, and released. The pendulum bob then undergoes simple harmonic motion with period T. The total mechanical energy of the pendulum bob is E. At the lowest point in its swing, the magnitude of the pendulum bob s velocity is v 0. 11. If the length of the string had been slightly longer than l, then, compared with T, the period of the motion would have been 3
12. If the mass of the pendulum bob had been slightly larger than m, then, compared with T, the period of the motion would have been 13. If the angle which the string makes with the vertical when the bob was released from rest had been slightly less than θ, then, compared to T, the period of the motion would have been 14. If the same pendulum was set in motion in the same way on the surface of the moon, then, compared with T, the period of the motion would be 15. If the length of the string had been slightly longer than l, then, compared with E, the total mechanical energy of the pendulum bob would have been 16. If the mass of the pendulum bob had been slightly larger than m, then, compared with E, the total mechanical energy of the pendulum bob would have been 4
17. If the angle which the string makes with the vertical when the bob was released from rest had been slightly less than θ, then, compared with E, the total mechanical energy of the pendulum bob would have been 18. If the same pendulum was set in motion in the same way on the surface of the moon, then, compared with E, the total mechanical energy of the pendulum bob would have been 19. If the length of the string had been slightly longer than l, then, compared with v 0, the magnitude of the pendulum bob s velocity as it passed through the lowest point in its swing would have been 20. If the mass number of the pendulum bob had been slightly larger than m, then, compared with v 0, the magnitude of the pendulum bob s velocity as it passed through the lowest point in its swing would have been 5
21. If the angle which the string makes with the vertical when the bob was released from rest had been slightly less than θ, then, compared with v 0, the magnitude of the pendulum bob s velocity as it passed through the lowest point in its swing would have been 22. If the same pendulum was set in motion in the same way on the surface of the moon, then, compared with v 0, the magnitude of the pendulum bob s velocity as it passed through the lowest point in its swing would be 23. Suppose an object with mass number m is attached to the end of a spring with requires a force F to stretch it a length l longer than it s relaxed length. What is the period of the simple harmonic motion of this system? 24. A flat board lying horizontally oscillates harmonically with an amplitude s 0. A book with mass number m lies on the board s rough surface. The coefficient of static friction between book and board is µ s. What is the maximum frequency the board can oscillate such that the book will not slip? 25. What is the frequency of small oscillations of a simple pendulum consisting of a plumb bob (mass number m) hung on the end of a string of length l? 26. If the period of oscillation of a simple pendulum is twice as large on the planet X as it is on earth (the same pendulum in both cases), what is the acceleration due to gravity on X ( g X ) compared to that on earth ( g e )? 6