# Spring Pendulum as a Simple Harmonic Oscillator Generating Sine Waves

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### The Idea

The simplicity of the spring pendulum provides an excellent opportunity to observe its motion in detail. The movement, like other vibrations in nature, follows a sine wave. We can also identify particular points in the spring's movement, such as where the velocity is at a maximum and a minimum during its cycle. We can also monitor how the force varies and how it relates to the acceleration. These relationships form the basis for a more complete understanding of how the various aspects of motion are interrelated.

### What You Need

• spring pendulum—set up as in previous experiments

### Method

1. Set the pendulum in motion and first observe when the following occurs in the cycle:
• Zero velocity
• Maximum velocity
• Zero force
• Maximum force
• Zero acceleration
• Maximum acceleration
2. Place a motion sensor to view the motion of the spring pendulum from underneath. If the mass presents a small target, you can tape an index card to the bottom of the mass to make it easier for the motion sensor to find. (To avoid air resistance, keep it small.)
3. Adjust the settings in the DataStudio program to give the maximum number of readings per second.
4. Open files to read simultaneously: distance, velocity, and acceleration.
5. Displace the spring and be ready to release it.
6. Press Start on the DataStudio screen to begin logging data.
7. Release the spring.
8. Record a few cycles.
9. Adjust the scales, if necessary, to best display the charts. Use the smoothing tools, if needed, to give a smoother curve if the acceleration data appears slightly choppy.

### Expected Results

The equilibrium position is the point where the stationary mass hangs without moving. At the equilibrium position, the velocity is maximum, but the force (and, therefore, the acceleration) is zero.

At the maximum displacement position (the point from where the spring was released), the velocity is zero and the force (and, therefore, the acceleration) is maximum. Graphs generated by a motion sensor measuring a pendulum are shown in Figure 68-1.

The distance versus time graph is a sine wave.

The velocity versus time graph is a cosine wave. The velocity is zero when the distance is at a maximum. The two waves have a similar shape, but the velocity curve is delayed by one-quarter of a period compared to the distance curve.

The acceleration curve is also a sine wave. It is at a minimum when the distance is maximum. The acceleration curve is zero when the distance curve is zero. The distance and acceleration curves have a similar shape, except the acceleration curve is delayed by one-half of a wavelength.

### Why It Works

A pendulum works because the further the mass moves from equilibrium, the greater the force that returns it to equilibrium. This is the basis of any uniformly vibrating object (known as a simple harmonic oscillator). The response of a restoring force, such as exerted by a spring, is to produce motion that follows a sine wave. The acceleration moves in the opposite direction as the distance because the force exerted by a spring is opposite its displacement from equilibrium. This also causes the velocity and acceleration curves to be out of phase with respect to the distance.

### Other Things to Try

A variation on this is to attach a force gauge to the spring to track the force along with the motion of the pendulum.

Physics alert: Those of you familiar with calculus will recognize that velocity is the first derivative of distance. Acceleration is the first derivative of velocity and the second derivative of the distance with respect to time. If the distance follows a sine curve, the velocity (the first derivative) is a cosine curve and the acceleration is a sine curve.

### The Point

A spring is a simple harmonic oscillator whose distance follows a sine wave pattern. Velocity and acceleration follow a similar shape, but are delayed with respect to the distance curve. The velocity is at a maximum at the point of thegreatest displacement. Acceleration is at a maximum at the point of greatest extension.