# Simple Harmonic Motion

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Lab 11: Simple Harmonic Motion

1. Purpose

The object of this experiment is to become familiar with Hooke’s Law and the properties of simple harmonic motion for a mass on a spring and a simple pendulum. For the mass on the spring, you will

1. A) Determine if the period depends on the amplitude of the oscillation
2. B) Find the spring constant of a spring using Hooke’s Law
3. C) Find the spring constant of a spring from the period of the oscillation

For the simple pendulum, you will

1. A) Determine if the period depends on the length of the pendulum
2. B) Determine the acceleration due to gravity by measuring the period of the oscillation
3. Background and Theory

Applying a force to the end of the spring will stretch (or compress) the spring. The larger the force that is applied the more the spring will stretch (or compress). The force the spring pulls back varies the farther the spring moves from the unstrained length. Empirically, it has been found the spring force is given by

Where the negative sign means the force is opposite the direction the spring is from equilibrium. k is called the spring constant of the spring.

If a mass is attached to the end of the spring and the spring is pulled away from the equilibrium position, the mass will oscillate back in forth. The force of the spring tries to bring the mass back to the equilibrium and is referred to as a restoring force. When the mass returns to the equilibrium position, it is moving. Its inertia causes it to move past the equilibrium position. The spring force again tries to return the mass to equilibrium. The motion continues. Since the restoring force depends directly on the displacement from the equilibrium, the motion is called Simple Harmonic Motion. The period is the time it takes for the motion to complete one repetition.

A simple pendulum consists of a mass attached to a point of support. The restoring force for the pendulum is a component of the gravitational force, mg sinθ. Pulling the mass away from the equilibrium position and releasing causes the mass to oscillate. For small angles, the sin θ is equal to θ and the restoring force is proportional to the distance from equilibrium.

III. Pre-lab Questions

As the mass placed on a spring increases, what happens to the period of the motion?

As we know that  . So as the mass increases so as the period of the motion increases keeping the spring constant same.

As the length of a pendulum increases, what happens to the period of the motion?

In case of pendulum  . So as the length of the pendulum increases, its period of motion also increases because T is directly proportional to the root of length. This implies that the period of motion increases.

1. Materials and Apparatus

Spring Meter stick

Simple Pendulum Hook Masses

Braces and rod Pendulum Holder

Motion Sensor 850 Interface

USB cable AC Adapter

Laptop

1. Procedure

Part 1: Amplitude Dependence of the Period for a Mass on a Spring

Setup Spring. Attach a large C clamp to the end of the table. Setup the rod in the clamp and the pendulum holder as shown. Turn one of the knobs on the pendulum holder and place the spring on behind the knob. Tighten the knob to secure the pendulum.

Setup the Motion Sensor. Connect the Motion Sensor into PASPort1 1 on the 850 interface. Connect the AC adapter to the interface and plug into the wall. Connect the USB cable to the back of the interface and to a USB port on the laptop. Place the Motion Sensor with the sensor facing up on the table underneath the spring. Turn on the interface.

Open the file. In the Physics 131 Labs folder, open the file Lab 11 Simple Harmonic Motion. If the Record button is grey, make sure the interface is connected correctly and turned on. Make sure the interface is on and the USB cable is plugged into a port on the laptop.

Check the sensor. Place a 100 g mass on the spring. Pull the mass down about 2.0 cm. Release the mass and click on Record. Remember to get out of the way the sensor. Check that the sensor is detecting the mass. The graph should be smooth. If it is, continue to the next step. If not, adjust the detector and try again. The mass should not get closer than 15 cm to the motion sensor. If it does, raise the pendulum holder higher.

Get the period. On the graph, highlight the data where the mass oscillates back and forth by clicking the highlight icon. A box will appear on the screen. Click the center of the box to move it to the data. Use the circles on the border of the box to change the size. Click on the down arrow of the fit icon and select the sine fit : Acos(ωt+φ)+C. A is the amplitude and the period can be calculated from ω by T=(2π/ω).

Change the amplitude. Repeat the experiment with amplitudes of 4.0 cm (between 0.035 and 0.045) and 6.0 cm (between 0.055 and 0.065).

From the data, does the period depend on the amplitude of the motion (are the periods different enough to say that the period does depend on the amplitude?)?

Part 2: Finding Spring Constant from Hooke’s Law

Measure the original length of the spring. Remove the 100g mass from the previous experiment. Measure the length of the spring from the top of the spring to the bottom of the spring. Record in the data table.

Place a mass on the spring. Start with a 50 g mass on the spring. Measure the new length of the spring. Record in the data table.

Change the mass. Repeat the experiment using 100g, 150g, 200g and 250g on the spring.

 Spring Constant from Hooke’s Law Original length of the spring Hanging mass (kg) Force on the Spring (N) Length of Spring (m) Stretch of the spring (m) 0.05 0.4902 0.6 0.100 0.9804 0.71 0.150 1.4706 0.83 0.200 1.9608 0.96 0.250 2.451 1.075

Using Excel, make a scatter graph of the Force on the spring (y-axis) vs the stretch of the spring (x-axis). Overlay a linear Trendline and display the equation.

The slope is the spring constant.

Part 3: Finding the Spring Constant from Simple Harmonic Motion.

Get the period of the motion. Switch back to the Lab 11 page. Using the same 5 masses as in Part II, put the mass on the spring, pull the mass 2.0 cm downward and release. Click on start. Obtain the period from the graph using the Sine Fit as before.

 Spring Constant from Simple Harmonic Motion Hanging Mass (kg) Period (s) Period squared () 0.050 0.64 0.100 0.78 0.150 1.04 0.200 1.21 0.250 1.43

Using Excel make a graph of the Period Squared (y-axis) vs Hanging Mass (x-axis). Overlay a linear Trendline and display the equation.

Write down the slope.

Calculate the percent error between the values of the spring constant found in part 2 and 3. use the spring constant from Part 2 in the denominator of the percent error equation.

As the mass increases, what happened to the period of the motion? Does this agree with your answer to the pre-lab question 1?

Part 4: Length Dependence of the Period of a Simple Pendulum

Setup the Motion Sensor. Remove the spring from the pendulum holder. Put the motion sensor on a stand by sliding the sensor onto the stand through the hole in the base of the sensor. Tighten to keep in place.

Measure the length of the pendulum. Place the pendulum string behind one of the knobs and tighten. The length of the pendulum is from where the string is pinned by the knob to the center of the ball. Make the length of the pendulum 20 cm. Record the length in the data table. Move the stand with the motion sensor so the motion sensor is pointing towards the pendulum. Adjust the height of the motion sensor until the ball is at the same height as the motion sensor. Make sure the ball is about 30 cm from the motion sensor.

Run the experiment. Pull the ball directly away from the motion sensor to a small angle and release. Click on Start. Make sure the graphs are smooth as before. Get the period from the sine fit as before.

Change the length of the pendulum. Repeat the experiment 4 more times increasing the length of the pendulum 5 cm each time.

 Length Dependence of Period for a Simple Pendulum Length of Pendulum (m) Period (s) Period squared () 0.20 0.8989 0.30 1.10 0.50 1.42 0.70 1.68 0.80 1.80

As the length of the pendulum increases, what happens to the period of the motion? Does this agree with your answer to the pre-lab question 2?

As the length of the pendulum increases, the period of the motion increases as well. This agrees with the answer to the pre-lab question 2.

1. Make a graph of the Period Squared (y-axis) vs Length of Pendulum (xaxis) Overlay a linear Trendline and display the equation.
2. Calculate the percent error between your value of g and 9.80 m/s2 with 9.80 m/s2 in the denominator of the percent error formula.
3. Close the program and select discard when asked to save changes.