Lab {10} Simple Harmonic Oscillation

Abstract
For the final laboratory in Physics I, student explored the motions of a mass on a spring. More specifically students looked at oscillations (periods) with various masses. Students set a spring into vertical oscillations with suspended masses and measured the period of oscillation. Using this method, students found a spring constant of 30.30N/m. Results should have verified that the period of oscillation depended on the effective mass of the spring and the period of oscillation. Students recorded basic information such as the position of the mass before the spring is charged, the path of the mass, the peak of the oscillation, as well as the force the mass and the spring exert on each other. Data studio and a force sensor, and a position sensor was used to get accurate measurements of these values.

Goal
The purpose of this laboratory is to characterize the oscillation of a simple spring-mass system.
Theory
If a spring is stretched or compressed a small distance from its equilibrium position, the spring will exert a force on the body given by Hooke's Law, where is known as the spring force. The constant, , known as the spring constant, and is the displacement from the equilibrium position. The spring constant is a clue of the spring's strength. A large value for indicates that the spring is strong or stiff. A low value for means the spring is weak or flexible. Springs with large values can balance larger forces than springs with low values. The negative sign in indicates that the direction of is always opposite the direction of the displacement. The spring force is a restoring force. In other words, the spring force always acts to return the weight or body to the equilibrium position regardless of the direction of the displacement, as shown in Figures A-C

Figure A The displacement is to the right
( > 0) when the spring force is directed to the left ( < 0). Figure C When the displacement is to the left ( < 0) the spring force is directed to the right ( > 0).

Figure C. Figures A and B, shoe the effect of the spring force is to return the system to the equilibrium position. At this position, = 0 and the spring is unstretched, signifying = 0.
When a mass, , is suspended from a spring and the system is allowed to reach equilibrium, as shown in Figure D, Newton's Second Law says that the magnitude of the spring force equals the weight of the body, . If the mass of a body is at equilibrium then the spring force acting on the body can be determined.

Figure D. A body of mass, , is suspended from a spring having a spring constant, . If the spring and body are in equilibrium the spring force is balanced by the weight of the body.

relates to a spring that is primarily unstretched. When the body experiences an random displacement from some initial position, , to some final position, , this equation can be written as

where is the body's displacement. If the body is displaced from its equilibrium position some maximum distance, Z, and then released, it will oscillate about the equilibrium position. The body will move back and forth between the positions x= + Z and x= - Z. When the mass travels from the maximum displacement x= + Z to the minimum displacement x= - Z and then back to the position x= + Z , the mass has moved through one oscillation. When an oscillating mass such as a weight bouncing on a spring experiences a force that is linearly proportional to its displacement but in the opposite direction which is called simple harmonic motion. This simple harmonic motion is periodic, which means the displacement, velocity and acceleration all vary sinusoidally. The time required for the body to complete one oscillation is defined as the period, , and is given by The period is dependent only upon the mass of the oscillating body and the spring constant, . As the k of the spring increases the period decreases which has the effect of increasing the body's average velocity. An increase in the body's mass means the period will also increase, thereby requiring more time for the body to move through one oscillation

Apparatus
 motion sensor
 scientific workshop interface
 iMac computer
 time-velocity experiments software
 weights (100g to 350g)
 hanging spring
Procedures
Part A: determining the spring constant k as demonstrated in lab 6.
Part B: determining the period of harmonic oscillation
1. Connection motion sensor, scientificworkshop interface, imac computer. Turn on.
2. Open time-velocity experiments.
3. Hanging 100g mass to the spring. And pull down the mass to start the oscillation motion
4. Start computer measurements.
5. From the graph and data table, compute the average nearest pick-to-pick elapse time. Record it as period T.
6. Repeat step 3-5 for 5 time with incremental 50g mass added to the hanger.

Data Table I. Spring constant k
Trial Mass Position k
1 0.000 0.000
2 0.050 0.014 35.035714
3 0.100 0.033 29.727273
4 0.150 0.051 28.852941
5 0.200 0.071 27.633803
Average 30.312433

Table II. Spring-Mass System
Trial Mass Num. of Cycles Total Time Period(Exp) Period(Theory) Error Period^2
1 0.150 10 4.86 0.5085 0.4420 15.047% 0.2586
2 0.200 10 5.97 0.5532 0.5104 8.392% 0.3060
3 0.250 10 5.23 0.6433 0.5706 12.739% 0.4138
4 0.300 10 7.06 0.7152 0.6251 14.419% 0.5115
5 0.350 10 6.74 0.7489 0.6752 10.923% 0.5609

Graph I Graph II Calculations for Table I Spring Constant (k)

• [(0.050-.000kg) * 9.81] / (0.014-.000m) = 35.03571429
• [(0.100-.000kg) * 9.81] / (0.033-.000m) = 29.72727273
• [(0.150-.000kg) * 9.81] / (0.051-.000m) = 28.85294118
• [(0.200-.000kg) * 9.81] / (0.071-.000m) = 27.63380282
• Average k value = 30.31243275

*Table II is reflected in Graph I ( Mass vs. Period) and Graph II ( Mass vs. Period squared)

Analysis and Discussion Graph I above show the mass (m) versus the period (T) and Graph II show the mass (m) versus the period squared(T2). In this experiment the students change the mass of the weight that were hung from the spring, m, and for each value observed the value of T (period). When students graphed T2 versus mass it was hypothesized that straight line with slope 4π2/k and intercept 4π2m/k. The line is not perfectly straight however, the r2 value (linear regression) for both graphs is 98% meaning the points on the graphs are almost in a straight line with one another.

Conclusion In part A, a spring was hung vertically with a mass hanger attached to the lower end of the spring, and masses from 0g to 200g were added. The downward location of the spring was measured once it came to rest. In this configuration, two equal and opposite forces acted on the hanging mass: gravity directed downward and the spring’s elastic restoring force directed upward, in the opposite direction of displacement. Using Hooke’s Law (F = -kx), a spring constant was calculated for each measurement. The spring constants for each value of displacement in Table I were averaged and found to be 30.30N/m by students and 30.312 N/m by Excel program.
In part B in reviewing Graph I and II with Table II there is some disagreement. The percent error values in Table II average around 10%, which may be considered a high percent error. However within Graph I and II the linear regression line is 98% meaning the points on the graphs have a strong correlation. This difference may be due to the fact that students were asked to run the system with no weights for Table II, however a clear output was not produced so the student used 350 g to test the system. The 10% error and linear regression line could correlation in the sense that the students arithmetic was off in all 5 trials but close enough that the linear regression came to be 98%. Even though the graph lines are not perfectly straight and the percent error was an average 10% it can still provide a relationship between the period, mass, and the spring constant though .
Resources
Jamal, Shah. Shah Jamal's Online Classroom. n.d. http://ebd24.com/physics-math-solution-of-simple-harmonic-oscillation.html. 18 April 2013.
Ouyang, Lizhi. "Tennessee State University." 31 August 2009. Physics. http://faculty.tnstate.edu/louyang/. 7 Febuary 2013.