# Simple Harmonic motion: The Spring Pendulum.

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### The Idea

A mass hanging from a spring is another example of a system that moves in a repeatable and consistent way. This is called simple harmonic motion. This experiment is about finding what causes a spring pendulum to vibrate faster or slower.

### What You Need

• several masses that can be attached to a string (such as 20 g, 50 g, 100 g, 200 g)
• everal springs of varying stiffness—it should be possible to partially stretch the spring by hanging each of the masses to the bottom of the spring. If the masses can't stretch the spring or if the spring is fully extended while supporting the mass, choose either other masses or other springs
• support for each pendulum
• stopwatch
• meterstick
• spring balance (for the extension)

### Method

1. Set up a spring pendulum consisting of a spring with one end supporting a mass and the other attached to a support above the spring.
2. Allow the weight of the mass to stretch the spring and come to rest.
3. Pull the pendulum straight down through a small displacement. (Increasing the elongation of the spring by about 10 percent is a good starting point.)
4. Release the pendulum and start the stopwatch as the pendulum is released. Try to release the spring, so it goes up and down in a vertical direction. Bear in mind that, after a few cycles, a spring may have a tendency to start swinging, which complicates the type of motion we are investigating here.
5. Count ten cycles up and down. Cycle number one is when the pendulum returns to its original position. Be careful not to count "one" when the pendulum is released.
6. Record the time (in seconds) for the pendulum to complete ten complete cycles.
7. Divide the time for ten cycles by ten to get the time for one cycle. This is the period of the pendulum for the conditions you are testing.
8. As with the previous study, you can approach this investigation in several ways. You are encouraged to develop your own approach to this. Here are a few suggestions:
• What variable matters: Mass? Spring stiffness? Amount of displacement? Test the selected variable, while holding the others constant. For instance, you can test squooshy, medium, and stiff springs, all using the same mass and displacement. (We define "squooshy" in quantitative terms in a minute.) Similarly, you can test light, medium, and heavy mass to determine whether the period of the pendulum is dependent on mass.
• Once you determine which variable(s) affects how fast the pendulum swings, you can set up an experiment to measure how the period changes over a range of the variable you selected. The other variables should be kept constant.

### Expected Results

The behavior of the spring pendulum is quite different than the swinging (simple) pendulum studied in the previous project.

Two variables are important for a spring pendulum: mass and spring stiffness.

The heavier the mass, the longer the period. Also, the stiffer the spring, the shorter the period. The "springiness" of a spring is called the spring constant, which gives a numeric measure of how stiff a spring is.

Within a fairly broad range, it should not matter whether you pull the spring through a small displacement or a larger displacement.

The dependence of period on mass and spring constant is not linear.

### Why It Works

The equation for the period of a spring pendulum (in seconds) is given by:

where m is the mass (in kilograms) and k is the spring constant (in N/m). Notice the period varies as the square root of the mass and the inverse square root of the spring constant.

### Other Things to Try

Predict and measure the period of the spring pendulum. You can do this by first finding the spring constant using the method of Project 30. You find the spring constant, k, by measuring the displacement, x, of a spring (in m) resulting from a given force, F (in N), according to the equation:

k = –F/x

The negative sign reflects the fact that force and displacement are always in the opposite direction resulting in a positive value for k.

Once you have determined the spring constant, predict the period of the spring pendulum using:

(The period will be given in seconds if the force is entered in newtons and the displacement in meters to get k. The mass must be in kg. Remember 1000g = 1kg.) Once you've called your shots, set your pendulum in motion and compare your prediction with your measured result.

### The Point

The period of a spring pendulum increases as the square root of the mass. The period of a spring pendulum increases inversely with the square root of the spring constant.