Simple Harmonic Motion in a Spring-Mass System

Abstract

Many things in nature are periodic: the seasons of the year, the phases of the moon, the vibration of a violin string, and the beating of the human heart. In each of these cases, the events occur in repeated cycles, or periods. In this project you will investigate the periodic motion of a spring, using a mini Slinky®. Basic physics will then allow you to determine the Hooke's Law spring constant. Your analysis will also yield the effective mass of the spring, a factor that is important in real-world engineering applications.

Objective

In this science fair project you will investigate the mathematical relationship between the period (the number of seconds per bounce) of a spring and the load (mass) carried by the spring. Based on the data you collect, you will be able to derive the spring constant, as described in Hooke's Law, as well as the effective mass of the spring.

Credits

David Whyte, PhD, Science Buddies

Slinky® is a registered trademark of Poof-Slinky, Inc.

Microsoft, Microsoft Excel is a U.S. registered trademark of Microsoft Corporation.

Last edit date: 2013-01-10

Introduction

This project requires very simple materials to explore the physics of periodic motion. All you need is a mini Slinky® and some weights, such as small fishing sinkers. The period of the Slinky is the time it takes to go through one down-and-up cycle when it is hung vertically from one end. The spring with the weight is a simple harmonic oscillator, which is a system that follows Hooke's law. Hooke's law states that when the simple harmonic oscillator is displaced from its equilibrium position, it experiences a restoring force, F, proportional to the displacement, x, where k is a positive constant:

Hooke's Law: F = -kx

As you add weights to the spring, the period (or cycle time) changes. In this project, you will determine how adding more mass to the spring changes the period, T, and then graph this data to determine the spring constant, k, and the equivalent mass, m_e, of the spring. The equation that relates period to mass, M, is shown below:

• M is the load on the spring in kilograms (kg).
• k is the spring constant in units of Newtons/meter (N/m).
• T is the period in seconds (sec).

In an ideal spring-mass system, the load on the spring would just be the added weight. But real springs contribute some of their own weight to the load. That is why the Slinky bounces even when there is no weight added. So the equation can be modified to look like this:

In this equation, the total mass pulling down on the spring is actually comprised of two masses, the added weight, m, plus a fraction of the mass of the spring, which we will call the mass equivalent of the spring, m_e. Rearranging equation 2, will give you the form of the equation you will use later for graphing, so:

Based on this equation, if you graph the added mass, m vs. , you will be able to find the spring constant, k, and the mass equivalent, m_e, of the spring.

Terms and Concepts

To do this project, you should do research that enables you to understand the following terms and concepts:

• Simple harmonic oscillator
• Hooke's law
• Simple harmonic motion
• Physics of springs
• Spring constant

Questions

• How does adding mass change the period of a spring?

Bibliography

• The Hyperphysics website has helpful diagrams explaining simple harmonic motion:
  Nave, C.R. (2006). Simple Harmonic Motion. Retrieved March 15, 2008 from the Departments of Physics and Astronomy, Georgia State University website: http://hyperphysics.phy-astr.gsu.edu/Hbase/shm2.html#c2
• Here is a brief introduction to Hooke's Law:
  Krowne, A. (2005). Hooke's Law. Retrieved March 13, 2008 from http://planetphysics.org/encyclopedia/HookesLaw.html
• For more advanced students, this high school physics tutorial on Newton's second law of motion can help you understand how to convert units of mass (hanging from the spring) to units of force (mass × acceleration due to gravity):
  Henderson, T. (2004). Newton's Second Law. Retrieved March 13, 2008 from http://www.glenbrook.k12.il.us/gbssci/Phys/Class/newtlaws/u2l3a.html

Materials and Equipment

To do this project, you will need the following materials and equipment:

• Mini Slinky
• Weights to hang from the spring. Here are some tips:
  • Fishing sinkers work well since they have holes in them for attaching to the spring. You could also use hex nuts, or AAA batteries attached to the wire with tape.
  • You will need five identical items to get a spread of data for the graph. The total weight should be around 35 g, or approximately 1 ounce.
  • Depending on the weights you choose, you might need fine wire or string to attach the weights to the spring.
• A scale for measuring actual mass of weights used, accurate to +/- 1 gram. Use an electronic kitchen scale, a scale from your school lab, or a postal scale.
• Stopwatch, or clock with a second hand
• Lab notebook
• Graph paper

Experimental Procedure

1. Do your background research so that you are knowledgeable about the terms, concepts, and questions above. Be sure to record your data in your lab notebook as you go along.
2. Measure the mass of one of your weights, using the scale. If your scale does not measure small weights, you can weigh all five of your weights and divide by five. Then measure the mass of the spring.
3. Perform the following steps to collect your data:
   1. Hold one end of the spring in your hand and let it bounce gently down and then back up.
   2. Count the number of cycles the spring makes in 60 sec with no weight hanging from it.
   3. Hang one weight from the spring (using a fine wire or string, if needed).
   4. Count the number of cycles the spring goes through in 60 sec with the weight attached.
   5. Perform at least three trials for each weight.
   6. Repeat steps c-e for a series of different weights.
4. Keep track of your results in a data table like this one. Try using the program Microsoft Excel to make the tables and perform the calculations when you work through this project.
   
   

   Load (mass added to spring) (g)	Number of cycles per 60 sec			Average
   	Trial #1	Trial #2	Trial #3

   
   
   
5. Make another table like the one below to convert your raw data into numbers that can be used to determine the spring constant and spring's effective mass.
   
   

   A	B	C	D	E
   Added mass (kg)	Average # cycles in 60 sec (1/min)	f, the frequency, or cycles per second (1/sec)	T, the period of spring, or the time for each cycle (sec)	(sec 2)
   Convert to kilograms	From the table above	Divide "Average # cycles in 60 sec" in column B by 60	Reciprocal of cycles per second in column C (divide 1 by the numbers in column C)	Multiply value in column D by itself and divide by 4(pi)2

   
   
   
6. Make a graph with "Added mass," m, in kilograms, on the y-axis, and , in sec², on the x-axis. Use kilograms rather than grams so that the value of k is in units of N/m, which is equivalent to kg/sec². Usually you are instructed to graph the independent variable (mass in this case) on the x-axis and the measured parameter () on the y-axis. You should ignore this rule in this project since graphing m on the y-axis will let you read m_e from the y-intercept.

   This graph of m vs. has the same terms found in Equation 3:

   

   Let's look at the equation. It has a form similar to the equation of a straight line: y = ax + b, where a is the slope and b is the y-intercept. In fact, Equation 3 is an equation for a straight line, with slope equal to k, the spring constant, and y-intercept equal to the negative value of m_e. In other words, there is a linear relationship between m and (), so a graph of m vs. () will be a straight line with slope k and y-intercept -m_e. The reason you calculated was to be able to read the values of k and m_e from the graph of m vs. .

   

   How do you determine the slope of the line you have drawn? The slope is measured as change in m, divided by the change in , over the same range.

   slope = k = Δm/Δ()

   Δm/ Δ() can be read as "delta m over delta ", with the Greek letter for delta, Δ, indicating "change of." Make another table like the one below to find the slope. Pick points that are near the ends of the graph, rather than adjacent points.

   
   
   

   Added mass (kg)	(sec2)	Δy (kg)	Δx (sec2)	Δy/ Δx (kg/sec2)
   		Subtract one y value from another, larger y value	Subtract one x value from another, larger x value	This is the spring constant, k, in units of N/m (kg/sec2)

   
   
   

   Once you have determined the value of the spring constant, k, from the slope of the line, you're ready to determine the effective mass of the spring. To do this, extend the straight line until it intersects the vertical y-axis. The line will intersect the y-axis at -m_e (negative m_e). Based on theoretical considerations, the absolute value of m_e should be around one-third of the mass of the spring.

Variations

• Perform the project with different types and sizes of springs.
• For an experiment using a spring-based mechanical model of the human knee, see the Science Buddies project Deep Knee Bends: Measuring Knee Stress with a Mechanical Model.
• For a project to investigate Hooke's law and to determine the spring constant by an alternative procedure, see the Science Buddies project Applying Hooke's Law: Make Your Own Spring Scale.

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