Acceleration Due to Gravity

Objectives

Although you can look up the value of “g” in a textbook, the acceleration due to gravity is not constant over the surface of the earth. The standard textbook value of \(\Variable{g}{9.80665}{m/s^2}\) is used to define a standard weight for an object of specific mass.

In this lab, we will measure the acceleration due to gravity in two different ways. Both approaches are based on kinematics. If you measure with enough precision, you may be able to observe a difference between the acceleration due to gravity in Klamath Falls and the textbook value.

This lab consists of three parts:

  1. Gather data from the spark timer.
  2. Analyze and plot these data to determine the acceleration due to gravity.
  3. Use video analysis software to determine the acceleration due to gravity.

First, we will use a spark timer to record the position of a falling object as a function of time. From these data, we will compute the acceleration due to gravity. Then, we will use a camera and video analysis software to measure the acceleration of a falling object.

Strive for accuracy, and try to identify any sources of uncertainty or experimental error that may influence your results in this lab.

Record your setup, procedure, data, and observations in your lab notes. Keep detailed notes. Data for this lab will include data tables and plots. In your memo, summarize your findings and describe any conclusions you draw from them. Report the values of \(g\) that you compute and include any tables or graphs not in your notes.

If you need additional materials to carry out the experiments below or your own, just ask Dr. Kinder.

When you finish both measurements, please submit your results here: https://forms.gle/rNYrmBstv2m8u8o76

Gather Data from the Spark Timer

Dr. Kinder will explain how to use the spark timer. There is a possibility of electric shock if the device is not used properly. (The voltage is high, but the current is low, so a shock will only be painful — not deadly.) Before you collect and analyze data, discuss the following:

  • What do you expect the position and velocity plots to look like?
  • What might limit the precision of the value of g with this method?

Analyze Data from the Spark Timer

Your raw data this week will be the locations of points on a strip of paper. From these data, you will need to compute the velocity and acceleration, and to make plots of position and velocity. You can do this using a spreadsheet, a calculator, or by hand.

Plan your measurements and analysis before you start taking data and crunching numbers. What columns should you include in your table? What is the best way to measure the locations of the points? Should you use a ruler or a meter stick? What range of values will you need to display in your plots? Planning now can save a lot of time in the analysis.

Create a graph of your velocity data.

Make sure your notes and analysis includes the following:

  • A plot of position vs. time
  • A plot of velocity vs. time
  • A line of best fit for your velocity graph, with values of slope and intercept (including units)
  • Your best numerical estimate of “g” in Klamath Falls

To find the line of best fit, you can draw a line of best fit by hand using a ruler, use linear regression, use software like Excel or MATLAB, or write a computer program. Whatever method you use, be sure to plot the line of best fit along with your velocity data and report the slope and intercept.

Questions to consider (and answer in your notes or your memo):

  • Do your velocity data follow a linear relationship?
  • What is the physical meaning of the slope of your velocity graph? The intercept?

Video Analysis

For the second measurement of \(g\) in Klamath Falls, we will use computer analysis of video of a falling object. We will use Vernier Video Analysis, which you can download and install on your own laptop (the license key is posted on Canvas). This program can carry out a frame-by-frame analysis of the position of moving objects in a video recording.

Learn about the software. Explore the software on your own, and don’t hesitate to ask Dr. Kinder for directions or tips.

Make and analyze a video. Record the motion of a falling object, then use Video Analysis to analyze the footage. You can use your phone to record the video, or borrow a camera from the lab.

Estimate the acceleration due to gravity. Extract data from your video to make your best estimate of \(g\). It is often more accurate to fit a line to the velocity data than to read values from an acceleration plot.

Make sure your notes and analysis includes the following:

  • A description of your setup for making the video
  • A plot of position vs. time extracted from your video
  • A plot of velocity vs. time extracted from your video
  • Your best numerical estimate of “g” in Klamath Falls

Question to ponder:

  • What limits the precision of the value of \(g\) with this method?
  • If you had more time and resources, how would you set up an even better experiment?

Further Exploration

The following activities are not required. If you have the time and the inclination, you may carry out these investigations for extra credit. You are also free to use any of the materials available in the lab to carry out your own investigations. You need to keep a record of your work (setup, procedure, data, analysis) to get credit for your efforts.

Additional Measurements: Make yet another measurement of the acceleration due to gravity. Some possibilities include the following:

  • Measure the period of a pendulum. The period of a pendulum is the time it takes the pendulum to swing back and forth once. For small amplitude swings, it is given by \(T = 2\pi \sqrt{L/g}\). By accurately measuring the length and period of a pendulum, you can deduce the value of \(g\). Note: The “length” of a pendulum, \(L\), is the distance from the pivot point to the center of mass of the swinging object — not necessarily the length of the string you use.

  • Measure the velocity of cart rolling down a ramp. You can use the LoggerPro software and motion detector, a stopwatch, Video Analysis, or a method of your own design. For a cart on an incline, Newton’s Second Law predicts that the acceleration down the ramp will be \(a = g \, \sin \theta\).

  • Use Video Analysis to measure the acceleration of two different falling objects. Textbook equations imply they should be the same. Are they, to within the uncertainty of the experiment? (Explain what these uncertainties are.) If not, is the problem with the experiment or with the textbook? (Justify your choice.)

Numerical Analysis: An important part of this lab is determining the acceleration from a discrete set of data points. Use calculus to determine the error in the following approximations for the velocity, as a function of the time step \(\epsilon = \Delta t\):

\[ v_n = \dfrac{s_n - s_{n-1}}{\Delta t} \qquad \implies \qquad v(t) \approx \dfrac{s(t) - s(t-\epsilon)}{\epsilon} \]

\[ v_n = \dfrac{s_{n+1} - s_{n-1}}{2\Delta t} \qquad \implies \qquad v(t) \approx \dfrac{s(t+\epsilon) - s(t-\epsilon)}{2\epsilon} \]

For each approximation, use a Taylor series to approximate \(s(t \pm \epsilon)\) out to terms of order \(\mathcal{O}(\epsilon^3)\). Then, evaluate the fractions. The error of the approximation is the power of the first power of \(\epsilon\) in the resulting expression. For example, saying “the error is linear” means that the coefficient of \(\epsilon\) in the result is not zero; “the error is quadratic” means the coefficient of \(\epsilon\) is zero, but the coefficient of \(\epsilon^2\) is not; and so on. A higher-order error implies a more accurate approximation when \(\epsilon\) is small.

Helmert’s Equation: The acceleration due to gravity is not a universal constant. Read the handout on Helmert's Equation. Use Helmert’s equation to calculate the expected acceleration due to gravity in Klamath Falls, Oregon. Are your data precise enough to measure the deviation from the textbook value? Can Helmert’s equation explain any discrepancy between your data and the standard value?