Simple Harmonic Motion
Often time in nature, when an object undergoes a small displacement from equilibrium position, it undergoes oscillatory motion about that equilibrium position; such behavior is called harmonic motion. Examples of harmonic motion include springs, pendula, and even circular motion. When no damping forces affect the oscillating object, the behavior is know as simple harmonic motion.
In this experiment, we will consider the motion of a mass attached to a vertical spring. The spring force that is on the mass is proportional to the masses displacement away from the equilibrium position , but opposite to the direction of the displacement :
Fs = – Kx
Here x is the displacement from equilibrium and k is the spring constant. The simple relation is know as Hooke’s Law: when the mass is displaced from its equilibrium position, it oscillates about that equilibrium point.
The time to make one complete cycle of oscillation is known as the period. For a mass on a soring, the period is given by
Where k is the spring constant on the spring and m is the mass of the object hanging from the spring. The spring constant measures the “stiffness” of the spring: a higher k means a stiffer spring.
Measuring the spring constant
When a mass is attached to a vertical spring , the mass stretches the spring some distance x but then settles at a new equilibrium position. At this point, the two forces are balanced, i.e.
Fs = Fg
So
Kx = mg
K = mg/x
In addition, the spring constant, K, can be determined by calculating the slope of the graph. The linear aspect of slope represents the spring constant as a constant feature. In an ideal case, regardless of the stretch the spring experiences, the spring constant remains the same. The slope of a Fs vs X graph gives the spring constant. The quality of linearity describes how well the spring constant remains constant during the experiment.
Figure 1: Example of a Fs vs X graph. The slope of the graph is 10 N/m which represents the K constant of the spring. The R2 suggests that the spring constant is constant during experimentation.
Procedure: Part 1 – Hookes Law & Spring Constant
Go to the following website: https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html and select INTRO.
⦁ In the gray check box on the right, Select
⦁ Natural Length
⦁ Equilibrium Position
⦁ Gravity = Earth
⦁ Use the slider and select any value to set the spring constant. You may make it as small or large as you want. Note the notch (0-10) your left the slider on in your data table. For example, this would be a “9”.
⦁ Hang a 50g mass onto the spring.
⦁ Using the ruler located on the right, measure the distance between the green lines and the blue lines. Record the measurement in meters.
⦁ Repeat with the 100g, and the 250g mass.
⦁ Pick the yellow unknown mass and measure the displacement and note it in the data table.
⦁ Convert the mass from grams to kilograms. 1000g = 1kg
⦁ Calculate the Fg = mg. Since this is in equilibrium with the Spring Force, we can assume that the magnitude of the gravitational force is the value for the spring force.
⦁ Graph the three known masses with Fs in Newtons on the y axis and the stretch on the x-axis in meters.
⦁ Show equation and R2 of the graph.
⦁ Determine the K constant by noting the slope of the graph.
⦁ Using your K constant, determine the mass of the unknown.
⦁ Repeat this entire process with another spring constant value and select the same unknown mass.
Analysis:
⦁ Calculate the percent difference between the two masses. Do they agree according to their percent different? Why or Why not?
⦁ Do the springs follows Hooke’s Law? Describe your reasoning with evidence.