1. What is the spring constant of a spring that starts 10.0 cm long and extends to 11.4 cm with a 300 g mass hanging from it?[br]2. List three places besides in springs where Hooke's law applies.[br]3. Show that [math]x\left(t\right)=A\cos(\omega t)+B\sin(\omega t)[/math] is a solution to the differential equation of the mass/spring system.[br]4. Does the period of simple harmonic motion depend on amplitude? [br]5. On what does the period of a mass on a spring depend?[br]6. Is the true pendulum an example of SHM? Explain.[br]7. What is the point of using a Taylor series in the context of the pendulum differential equation?[br]8. What period do you expect to find for small amplitude swinging motion of a rock tied to a 1.0 m long string?[br]9. Using the animated slider in the book, estimate the period of the same rock caused to swing at a 90 degree angle from vertical. NOTE: It is possible to drag and zoom the plot in the book to get a precise answer.[br]10. Why does the actual pendulum's plot of angle vs time flatten out at very large swing angles? Give a clear physical explanation.[br]11. At what point in SHM is the velocity maximum?  Displacement maximum?[br]12. What could we conclude if a system has a phase trajectory that sweeps out larger and larger area as time goes by?[br]13. What does a repetitive phase trajectory indicate about a system?[br]14. What does each point in phase space represent?[br]15. Given an oscillator of mass 2.0kg and spring constant of 180N/m, what is the period without damping?  Use numerical methods to model this oscillator with an additional friction force equal to [math]F_{damping}=-cv[/math] where c is a positive damping constant.  Using c=5.0, what is the new period of oscillation.  What about for c=10? Assume initial position is 0.2m and initial velocity is zero. Please find the period using the position versus time plot and use the first full cycle of the motion.[br]16. Critical damping is the case where the mass never actually crosses over equilibrium position, but reaches equilibrium as fast as possible.  Experiment with changing c to find the critical damping constant.  Use the same initial conditions as in the last problem.  Zoom in a bit to make sure you don't allow any oscillations to take place - even small ones.[br]17.  Shocks on cars are usually designed to achieve critical damping of the suspension system when the car is loaded with maximum number of passengers and cargo.  With only a driver and minimum cargo, is the car over-damped or under-damped?  Over-damped means even more damping than the critical amount and under means the opposite.[br]18. A 2.0kg mass on a spring with elastic constant 32 N/m starts at a position of x=0.3m away from equilibrium with a velocity of 0.4 m/s. What will its maximum displacement be? Maximum velocity? Maximum acceleration? What will those three values be at t=5.0s?[br]19. A pendulum length is doubled. What happens to its period?[br]20. A mass hanging from a pendulum is doubled. What happens to the period?

1. 214 N/m[br]2. molecular bonds, bending rods, twisting objects[br]3. see chapter[br]4. no. otherwise inside the sine function would be the amplitude.[br]5. spring constant and mass[br]6. no. it is not a linear restoring force, has a period depending on amplitude and is not solvable analytically in terms of sines and cosines.[br]7. If we approximate the sine function with the first term in the series, we are able to solve the differential equation analytically for small angles.[br]8. 2.0s[br]9. approximately 2.4s[br]10. because the restoring force decreases at larger angles and goes to zero at [math]\pi[/math] radians.[br]11. at zero displacement, at zero velocity.[br]12. energy is being added by some outside source.[br]13. that it's not losing energy[br]14. a unique state of the system.[br]15. 0.661s, 0.671s, 0.691s[br]16. [math]37.9 kg\cdot m^{1/2}/s[/math][br]17. over-damped[br]18. 0.32m, 1.3m/s, 5.1m/s[sup]2[/sup], 0.21m, -0.93m/s, -3.42m/s[sup]2[br][/sup]19. [math]\sqrt{2}[/math] longer duration[br]20. remains the same