Hooke’s Law and the Simple Harmonic Motion of a Spring

Hooke’s Law and the Simple Harmonic Motion of a Spring
Purpose:  To determine the force constant of a spring and to study the motion of a spring and mass
when vibrating under influence of gravity.
Equipment: Spring, masses, weight hanger, meter stick, support stand with clamps, motion detector,
LabPro interface, wire basket.
Introduction: When a spring is stretched a distance x from its equilibrium position, it will exert a restoring
force directly proportional to this distance. We write this restoring force, F, as:
1)  F  =  -kx      
where k is the spring constant and depends on the stiffness of the spring.  The minus
sign remind us that the direction of the force is opposite to the displacement.
Equation 1 is valid for most springs and is called Hooke’s Law.
If a mass is attached to a spring that is hung vertically, and the mass is pulled down
and released, the spring and the mass will oscillate about the original point of
equilibrium. Using Newton’s second law and some calculus we can show that the
motion is periodic (repeats itself over and over) and has period, T (in sec), given by
 2)  T  =
4
   
where m is the mass supported by the spring.
table
table clamp
spring
hanging
mass
support rods
π
2
k
m
Procedure:
1. Hang the spring on the support rod, as shown in the diagram, and measure the position of the lower end                                                      
of the spring. Place 350 gm mass on the spring and observe its position. Now attach, in turn, masses of
450, 550, 650, 750, 850, 950, 1050 gm and measure how far the spring is stretched for each of these
masses.
2. Make a plot of the downward force applied to the spring (y-axis) versus the displacement of the spring (xaxis). Remembering Equation 1, determine the force constant, k.
3. Start up the Logger Pro software by clicking on its icon in the Physics Apps folder.  From the program
click on Open/Mechanics/Hooke’s Law  to open the file for this experiment.  A graph of linear position
vs time should appear.  Place the motion detector on the floor beneath the hanging mass.  Place the wire
basket over the motion detector to protect it from any accidentally dropped masses.  Once again place the
350 gm mass on the spring and pull the mass downward until the spring has been stretched 10 cm.
Release the mass and observe the subsequent motion. Start collecting data by clicking on the Collect
button. The time scale on your graph should allow for at least five cycles of the motion to be seen.  Press
the “x=” button and determine the time for five cycles. From this number calculate the period of motion.
Record your data.  Repeat this for the other masses used in part 1.  Create a data table which gives
average T and m values.
4. Using the data for the last trial (with m = 1050 g), fit the data to a sinusoidal function using the
Analyze/Curve Fit option.  Determine the period and the amplitude from your function.  Compare the
period with the value obtained in part 3.
5. Make a plot of T2 (y-axis) vs m (x-axis).

Conclusion:

The whole point of Hooke's Law in this experiment was to understand the conservation of momentum, potential gravitational energy, potential spring energy, and the conservation of energy with regards to projectile motion. Since Hooke's Law is F=-kx where k is the spring constant and x is the displacement of the spring due to compression or extension, it's understood that the force is in the opposite direction of the displacement because of the signs in the equation.