**Our scenario** deals with a mass that hangs at the end of a spring. We assume the spring is attached to the ceiling. The weight can be given a push so it starts to move. When the spring is in motion, we describe it’s position relative to equilibrium.

**Constructing a Simple Model**

we’ll assume that we have a weight hanging from a spring that is perfectly elastic. We’ll also ignore other variables such as air resistance and the mass of the weight at the end of the spring.

To start, we assume that the weight will always move at most **20 centimeters** (cm) above or below equilibrium no matter where it starts. In addition, it takes **4 seconds to make one full cycle**.

**You’ll need to construct and graph a function h(t) that models the height of the weight above (and below) equilibrium in each of the following scenarios. In each case, graph the model over one full period starting at t = 0. For the ï¬rst four models, you will need to construct a model of the form Asin(bt) or Acos(bt). For the next two models, you’ll need to ï¬nd a function of the form Asin(bt + c) or Acos(bt + c).**

**1. Initial push is upward from the equilibrium point.**

**2. Initial push is downward from the equilibrium point.**

**3. The weight is pulled 20 cm above equilibrium, and the initial movement at t = 0 is downward.**

**4. The weight is pulled 20 cm below equilibrium, and the initial movement at t = 0 is upward.**

**5. The weight is pulled 10 cm above equilibrium, and the initial movement at t = 0 is downward. For this model, you’ll need to ï¬nd the value of c using the information about the spring and weight at the starting time.**

**6. The weight is pulled 10 cm below equilibrium, and the initial movement at t = 0 is upward. For this model, you’ll need to ï¬nd the value of c using the information about the spring and weight at the starting time.**

**7. For models 5 and 6, what is the phase shift?**

**8. For the ï¬rst four models, at what times after the start of the experiment does the weight pass through the equilibrium position? At what times is the weight 20 cm above equilibrium? At what times is it 20 cm below equilibrium?**

**9. For models 5 and 6, suppose we set oï¬€ two separate springs based individually on these models, both at time t = 0. At what times are these models both at the same position, and where is this position located?**

Subject | Mathematics |

Due By (Pacific Time) | 12/04/2015 12:00 am |

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