Mechanical Energy of a Spring - Warm up and Prediction

1. Make a sketch of the system before a mass is hung from the end of the spring. Draw a coordinate system, placing the origin at the end of the hanging spring when the spring is unstretched. Approximate the spring as massless.
2. Draw the spring system when a mass is hung from the spring and resting at its equilibrium position. Draw a force diagram for the object hanging at rest from the end of the spring and solve for the spring constant in terms of the object's mass, the gravitational acceleration constant, and the object's equilibrium position.
3. Consider the situation where an object, attached to the end of the spring, is initially held at the position at which the spring is unstretched and then released so that it begins oscillating. Make a sketch of this system when the object is at some arbitrary position. Determine the different kinds of energy found in the system while the object is oscillating.
4. What is the total energy of the system just before the object is released? How is this related to the other forms of energy in the system?
5. Assuming that total energy is conserved, although the values of the different forms of energy change during the oscillation, determine how the spring potential energy depends on the kinetic energy and the gravitational potential energy of the object and, in turn, how the kinetic and gravitational potential energies of the object depend on its position and speed.
6. Finally, write down the theoretical form for the spring potential energy. How could we plot the spring potential energy (as determined from the answer to problem 5) as a function of position to easily show that this theoretical form holds? Will a plot of spring potential energy versus position be linear? How could we adjust position or spring potential energy to make this plot linear? What would be the slope of this plot? (The section "Using Linear Relationships to Make Graphs Clear" in the appendix "A Review of GraphsDownload A Review of Graphs" will help you answer this question.)

Prediction

Knowing the value of spring potential energy, determine how it can be plotted versus some function of position to yield a linear plot, and determine the slope of this plot.

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