Pendulums The motion of a certain pendulum can be modeled by the function d = 4 cos πt where d is the horizontal displacement in inches relative to its position at rest and t is time in seconds. (a) Graph two cycles of the function. (b) What is the greatest horizontal distance the pendulum will travel from its position at rest? Buoy A buoy oscillates up and down as waves go past. The buoy moves a total of 3.5 feet from its low point to its high point, and then returns to its high point every 6 seconds. (a) Write an equation that gives the buoy’s vertical position y as a function of time t if the buoy is at its highest point when t = 0. (b) Explain why you chose y = a sin bt or y = a cos bt. (c) Graph 3 cycles of this equation. (d) What is the height of the buoy at t = 3 seconds? 15 seconds? Ferris Wheel Suppose you are riding a Ferris wheel for a total of 180 seconds. Your height h (in feet) above the ground at any time t (in seconds) can be modeled by the equations h = 87 sin (π/20 – π/2) + 90. (a) Graph your height above ground as a function of time for the entire 180 seconds. (b) What are your maximum and minimum heights during the ride? (c) How long does it take to make one loop? (d) Say the power goes out 150 seconds into the ride. How high off the ground are you? Respiratory Cycle For each person at rest, the velocity v (in liters per second) of air flow during a respiratory cycle is v = 0.85 sin π/3 t where t is the time ( in seconds). Inhalation occurs when v>0, and exhalation occurs when v<0 (a) Graph three cycles of this equation. (b) Find the time for one full respiratory cycle. (c) Find the number of cycles per minute. (d) Find the maximum velocity of the air flow. (e) How would the graph and equation change if the person was not at rest but exercising? Physics A weight on a spring exhibits repetitive motion. Imagine a weight hung from a ceiling by a spring. The system is in equilibrium when the weight is motionless. If the weight is pulled down or pushed up and released, it would tend to oscillate freely if there were no friction. In a certain spring-mass system, the weight is 5 feet from a 10-foot ceiling when it is at rest. The weight starts its motion 2 feet from the floor and moves towards the ceiling. It takes 2 seconds for the weight to return to a height of 2 feet off the floor. Let h be the height off the floor (floor will have h value of zero) and let t be time in seconds. (a) Draw a picture of this spring as it travels vertically. (b) Write an equation that represents this situation and graph 3 cycles of this equation. (c) How far was the weight from the ceiling when it was released? (d) How close will the weight come to the ceiling? (e) When does the weight first pass its equilibrium point? (f) What is the greatest distance that the weight will be from the ceiling? (g) Find the period of the motion. (h) Find the amplitude of the motion. (i) What is the frequency of the motion? (j) How far from the ceiling is the weight after 2.5 seconds?