# Simple Harmonic Motion

Familiarize yourself with the simulation. Play around with the settings, pull the pendulum to a side and let it swing, change the length, mass, gravity and friction; notice how each one effects the motion of the pendulum. When you are finished testing all the settings, click on the “Reset All” button.

Step 1: Set the length “L” of the pendulum equal to 1.0 m and mass of the bob equal to 0.1 kg. Keep the Gravity to “Earth” and the friction slider at “none”. Check the “Stopwatch” box, it will start displaying a stopwatch that you will use to measure the time period of the pendulum.

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Step 2: Pull the bob to a side, you will notice the angle that the pendulum makes with the vertical line appears at the top end of the pendulum, keep this angle 10 degrees or less for all the measurements below. Let the bob swing. Hold your mouse on “Play” button of the stopwatch, and observe the pendulum carefully. When the bob reaches to the extreme left end, start the stopwatch, and start counting the number of oscillations: when the bob starts at extreme left goes to right and comes back to the extreme left where it started, that is one oscillation. Click the “Stop” button on the stopwatch when the pendulum completes 10 oscillations. Record this time in the table below along with the corresponding length. Use a calculator to find the time for a single oscillation (also called time period), and record it in the table below. Click the reset button on the stopwatch.

Step 3: Keeping the bob mass fixed at 0.1 kg, repeat Step 2 for pendulum length of 0.9 m, 0.8 m, 0.7 m, 0.6 m, 0.5 m, 0.4 m, 0.3 m and 0.2 m. Record all the data in the table.

 Time for 10 oscillations T10 (s) Length L (m) Time period   (s) 1.0

Step 4: Open the Online Graph Maker at https://chart-studio.plotly.com/create/#/

Step 5: Use your mouse to select only the data in the last two columns of your table above (starting from 1.0, not including the heading row that has L and T), copy this data. Paste this data into the first two columns of the grid on your Online Graph Maker webpage clicking in the first cell of the grid in Column A and pressing ctrl+v on the keyboard.

Step 6: Click on “+Trace” button, and from the drop-down menu in front of X select “A”, from the from the drop-down menu in front of Y select B. So, it will make plat with data in column A (length L) of the grid along the x-axis and data from column B (time period T) along the y-axis. You will see the graph appear under the grid. Notice the trend on the graph, as the length of the pendulum increase, the time period also increases, but it is a curve not a straight line.

Step 7: Let’s fit the data to a mathematical function. From the left-hand side menu, click on “Analyze” and from the drop-down menu click on “Curve Fitting”. You will see a “+fit” button appear, click on it. From the drop-down menu in front of “Select A Trace”, select trace 0. Then click on the “Advanced” button. In the “Custom Function” box, replace the fit function m*x+b with M*x^B. Click anywhere outside this box. Under “Parameter Estimates” in front of M scroll up the arrow and select 1, similarly scroll up the arrow in front of B and select 1. Click on “Run” button at the bottom of this menu. You will see a fit curve appear that goes through the data on your graph. Notice that the M and B values under the “Parameter Estimates” have changed, these new values best fit your data, copy these values of M and B and past in the space below.

M (experiment) =

B (experiment) =

Take a screenshot of your graph along with the parameter estimates M and B, and paste it in the space below.

Step 8: Let’s find out how the time period depends on the gravity. Set the length of the pendulum equal to 1.0 m and mass of the bob equal to 0.1 kg. Select “Moon” from the drop-down menu under Gravity. Measure the time period for the pendulum on the Moon, using the method described above in Step 2. Record it below

TMoon =

What is the time period for the same pendulum on Earth from the data table on the previous page (length 1.0 m, mass 0.1 kg and gravity of Earth)?

TEarth =

Is the time period smaller or bigger on the moon compared to that on Earth?

What could be the reason that makes the time period on moon smaller (or bigger)?

Step 9: Let’s find out how the time period depends on the mass of the bob. Set the length of the pendulum equal to 1.0 m and mass of the bob equal to 0.5 kg. Measure the time period for the pendulum on the Moon. Repeat this step for masses of the bob equal to 1.0 kg and 1.5 kg respectively, keeping the length fixed at 1.0 m. Record your data below.

T (m = 0.5 kg) =

T (m = 1.0 kg) =

T (m = 1.5 kg) =

Does the time period increase or decrease when the mass increases?

Step 10: The time period for a pendulum is given by:

We already discovered the relationship between the time period and the length of the pendulum (keeping the gravity fixed) in Step 7 above. The equation we used to fit the data was:

Where “x” in Step 7 was the pendulum length L.

Comparing the two equations above, you would expect that:

B (theory) =

M =

Use the experimental value from Step 7 above and the actual value from theory to calculate the percent error in B. Show all your work below.

______________

Step 11: Now since:

M =

Put the value of constant M from Step 7 into the equation above and solve the equation for the gravity . Show all your work below.

M =

The actual value of the gravitational acceleration here on Earth is:

Step 12: Find the % error. Show all your work below:

______________

Step 13: Let’s find out the gravity on an unknown planet. Go back to the simulation, set the length of the pendulum equal to 1.0 m and mass of the bob equal to 0.1 kg. Select “Planet X” from the drop-down menu under Gravity. Measure the time period for the pendulum on Planet X. Record it below

TPlanet X =

Use this time period to find the gravity on Planet X. Show all your work below.

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