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2.1: 2.2: 2.3: 2.4: By drawing a free body diagram for a simple pendulum and breaking the forces into components, show that the restoring force on a pendulum obeys equation 3 if we ignore friction and air resistance. (Hint: the net restoring force for a pendulum is the component of the sum of the forces that is perpendicular to the pendulum's string.) Reconsider the section in the introduction concerning the small angle approximation. Based on equation 4, what is the effective \"Spring constant,\" k, of a pendulum for small angles of displacement? (Hint: compare equations 4 and 1.) For small angles, find an expression for the period of a pendulum. (Hint: compare the value of the \"spring constant\" you found in 2.2 and equation 2.) Open the pendulum simulation. And move to the "Intro\" tab. Set the length of the pendulum to 0.25 m, the gravity to \"Earth," and the mass to 0.5 kg. Make sure the friction is set to none. l J:..'I'I-.JI..|i.JI'l'I l tll' 2.5: Start the pendulum moving by displacing it by no more than 5 degrees, then use the stopwatch provided by the simulation (or one of your own) to Measure the period of the pendulum. Do so by measuring the time it takes 2.6: 2.7: 2.8: for the pendulum to return to its maximum ten times then divide that value by ten to get an experimental value for the period. Be sure to estimate your uncertainty. Compare your measured period to the period you calculate based on the equation you derived in 2.3. Repeat steps 3.5-3.6 for pendulum lengths of 0.5 m and 1 m. Reset the length of the pendulum to 0.75 m. Repeat steps 2.5-2.6 with masses of 0.5 kg, 1.0 kg, and 1.5 kg. How did the period vary as a function of length or mass in 2.7 and 2.8? Does this match what you expected based on the formula you derived in 2.3? How might you use a pendulum to measure the acceleration due to gravity on an unknown planet? Under what conditions is Hooke's Law invalid? Does a spring always exhibit simple harmonic motion? Does a pendulum? What is simple harmonic oscillation