Undamped Harmonic Motion

1. Hooke’s Law: The force exerted by a spring is proportional to the spring’s displacement from its rest position.

a) Write an equation for Hooke's Law where F is the force (measured in pounds in the problems below) and x is the displacement (measured in feet or inches in the problems below).

b) The proportionality constant you used in this equation is called the spring constant. If an object weighing 12 pounds stretches a spring 3 inches, what is the spring constant for this spring? (Include units.)

c) Do the springs in an "extra firm" mattress have a large spring constant or a small spring constant relative to a less firm mattress? Explain.

2. Simple Harmonic Motion: Simple harmonic motion, also known as free undamped motion, can be modeled by the ODE

                     (1)

where x is the displacement of the object from the rest position, t is time, and , where k is the spring constant and m is the mass of the object. We can solve to find

                    (2)

is the solution to (1).

a) To get a feel for this model, graph various (and interesting) values for c1, c2, and w and graph the curve x(t) for each choice. Discuss how the curve changes for different values of the parameters.

b) Define, illustrate, and show a calculation for period as it applies to simple harmonic motion. Be sure to include units in your definition.

c) Define, illustrate, and show a calculation for frequency as it applies to simple harmonic motion. Be sure to include units in your definition.

3. In this problem, you are to draw all solution curves by hand. Keep in mind that if an object is hanging from a spring, the positive values of x indicate that the spring has stretched. Therefore, positive values of x indicate that the object is lower than it was when the spring was at rest. Also, positive values of  indicate that the object has a velocity which stretches the spring. Therefore, an initial condition which has a positive value for  moves the object downward with an initial velocity.

a) Draw three periods of the solution curve to (1) where the object on the spring is pulled 3 inches below its resting position and then released. What are the initial conditions?

b) Draw three periods of the solution curve to (1) where the object on the spring is pulled 3 inches below its resting position and then released BUT the spring has a HIGHER spring constant than in question a). Use the same scale as in a) so the graphs can be compared.

c) Draw three periods of the solution curve to (1) where the object on the spring is pulled 3 inches below its resting position and then thrown downward with and initial velocity of 2 inches per minute.

d) Draw three periods of the solution curve to (1) where the spring is initially compressed by 2 inches and the object is initially thrown downward with a velocity of 2 inches per minute.




This lab was inspired from exercises in Differential Equations by Blanchard, Devaney, and Hall and question 1c comes directly from that text.