Chabot College

Physics Lab

Scott Hildreth

 

Determining the Acceleration Due to Gravity

 

Introduction

 

In this experiment, you’ll determine the acceleration due to earth’s gravitational force with three different free-fall methods. We say an object is in free fall when the only force acting on it is the earth’s gravitational force. No other forces can be acting; in particular, air resistance must be either absent or so small as to be ignored. When the object in free fall is near the surface of the earth, the gravitational force on it is nearly constant. As a result, an object in free fall accelerates downward at a constant rate. This acceleration is usually represented with the symbol g.

Your goals for the exercise are:

 

-       Explore the use of graphs and equations for distance vs. time, and velocity vs. time, to determine “g”, the earth’s gravitational acceleration.

-       Master the difference between precision - measuring carefully and accounting for uncertainty in your measurement - and accuracy - measuring correctly and coming close to the “right” answer.

-       Work together in a group setting, dividing up the work fairly and double-checking one another. You will need at least two people to take data, two people to draw graphs using Excel, someone to keep the team on task, and someone to ensure that the write up is completed and turned in on-time.

 

Write up Expectations

 

Turn in one full write up for the team, on or before Wednesday, 17 February. Include:

a)     the names and lab duties of each of the participants, and short abstract

b)    the organized data tables for all experiments

c)     the graphs you drew, and the calculations of the slope of the velocity vs. time curve that determine the value of “g”

d)    analysis of sources of errors and uncertainties in your work.

 

 

You are encouraged to work together using our on-line forum, http://clpccd.blackboard.com.

 

NOTES!

-          You may do these experiments in any order.

-          The value of “g” (actual) to use for comparison is 9.81 meters/sec2


Chabot College Physics – Hildreth

Determining the Acceleration of Gravity

 

 

Experiment 1: Simple Free-Fall

  

Procedure:

  1. Measure the height the ball falls from the edge of the table to the sensor pad on the floor. (Be sure to limit your measurement to the proper number of significant digits, and include your uncertainty as “+/-“; for example, 100.2 cm +/- 0.2 cm)

  2. Determine the % Uncertainty in your measurement. Percentage Uncertainty is given for any measurement by:

 

[Uncertainty]/[Measured Value] x 100%

For the example: [0.2 cm] / [100.2 cm] x 100% = 0.2%

  1. Time the fall using the electronic photogate setup. Be sure to estimate a reasonable uncertainty in your measurement of the time, as well, based on the photogate AND your reaction time.

  2. Repeat the experiment at least 10 times. Record every value of the time, even those that don’t work as well. Calculate the average time, and the average uncertainty for this time in this experiment, which will be about the the average uncertainty for any one measurement divided by the square root of “n” – the number of trials.

    This is the reason you take multiple measurements in an experiment – to minimize overall uncertainty, and increase your precision. Note that if you are taking the measurements incorrectly, or if the apparatus you are using is incorrectly calibrated, you still might get the wrong answer! So taking multiple measurements may not increase your accuracy!

    Now you can use the average time of the 10 measurements and the distance it fell to determine the acceleration “g”

  3. Determine the value of “g”. How?

    What You Know: (a) the distance the ball falls (displacement), (b) the time it takes, and (c) the initial velocity of the ball when it starts; vo= 0. If you know 3 things, you can find the fourth – the acceleration! What equation can you use?

  4. The overall uncertainty (dg ) in your measurement of g, using the percentage uncertainty in the measurements of time and distance, depends upon both of the uncertainties you estimated in time and distance:

Relative uncertainty in acceleration of gravity (dg/g) = 2 (dt/t) + (dy/y)

 

Note that the time element’s uncertainty appears multiplied by (2) because the equation for acceleration depends upon the SQUARE of the time. So the time variable appears “twice”, and each time its uncertainty matters.


6. (cont.)

The percentage uncertainty in “g” will then be an estimate of your overall precision for the experiment:

% Uncertainty in “g” = 100% x dg/g = 2 x (% uncertainty in time) + (% uncertainty in distance)

 

  1. Calculate the actual percentage error in your results; this is a measure of how accurately your experimental value of “g” compares with the actual value:

 

[“g”(actual) – “g”(experimental)] x 100% / (”g” actual)

 

Example: Your experimental value of “g” was 8.90 m/sec2; the actual is 9.81, and your actual percentage error would be:

 

[9.81 – 8.90] x 100% / (9.81) = 9.3%

 


Experiment 1 Data Table
 

 

 

 

 Drop Distance +/- ?

 

Estimated Uncertainty in Individual Drop Time:

 

Drop Times:

T1:

 

T2:

T3:

T4:

T5:

 

T6:

 

T7:

T8:

T9:

T10:

 

Average Drop Time:

 

 

 

% Uncertainty:

 

 

 

 

Average “g”

 

 

% Uncertainty

 

 

 

 

 

Experiment 1 Questions:

  1. When you measure the distance of fall, do you measure from the bottom of the ball to the top of the pad, or from the middle of the ball where the clamp holds it?

  2. What effect do you think the clamp has on the time of free fall? How can you estimate the significance of this effect, if any?

  3. What effect do you think human reaction time has in this experiment?

  4. Discuss the difference between precision and accuracy in this experiment. Compare your actual percentage error with the percentage uncertainty in your measurements.

Chabot College Physics – Hildreth

Determining the Acceleration of Gravity

 

Experiment 2: Timed Free-Fall

 

You’ll use a spark timer to make marks on a tape as an object falls. The timer will produce a spark every 1/60th of a second (+/- 0.005 seconds). We’ll do this experiment once, but analyze the resulting data in two different ways. A suggested data table for this experiment is attached, but you are free – and encouraged – to create your own.

Method 2A uses the total distance fallen and the total elapsed time to estimate an “average” velocity in each time interval, which can be used to create a graph of velocity vs. time, and from that, acceleration.

Method 2A Procedure:

1.      Each team will create their own free-fall data strip using the spark timer. The sparks will mark the tape at constant intervals of 1/60th of a second. Pick one spot at the top of the tape, label that spot “top”, and then mark the tape, numbering the spots from the first spot beneath the “top” to the bottom (from t1ti ).

 

  1. Measure the total displacement from the first spot to each spot, yi. In other words, go from the top spot to the first spot, and measure the distance (y1). Then go from the top spot to the second spot, and measure the distance (y2). Then from the top to the third spot down, and measure the distance, etc. Record these displacements in a second table within Excel or on paper.

  2. Calculate and record the overall average velocity of the falling block at each point, which depends upon the total distance traveled and the total time elapsed to fall that distance:

 

vi (avg) = 2yi/ ti

 

  1. Determine the value of “g” by graphing average velocity of the falling block vs. time, and measuring the slope. The value of the slope will approximate the value of “g”. (You can use Excel’s statistics functions to determine the value of the slope and its uncertainty statistically.)

 

The uncertainty in your value of “g” will be influenced both the average uncertainty in time, the average uncertainty in measurements of distances travelled, and the number “n” of data points you recorded. The uncertainty in time is used twice (once to get velocities, and again to get acceleration); the uncertainty in distance is used once. Note that “n” increase, your overall uncertainty decreases by the square root of n.

 

%dgA = [(2 x % Average Uncertainty in Time) + (Average % uncertainty in distance)]/ √n

  1. Record both the value for gravity and your uncertainty. gA +/- dgA %.

 

  1. Calculate the actual percentage error in your results to determine the accuracy of your experiments:

 

[“g”(actual) – “g”(experimental)] x 100% / [”g” actual]

 

 

 

Method 2B uses the displacements between each mark as the object falls, and the known time interval to estimate an “instantaneous” velocity for that interval, that will continue to increase as the object is accelerated downwards by the force of gravity. If you graph this instantaneous velocity at each point vs. time, you can estimate the slope of the line and therefore, the acceleration of gravity.

Method 2B Procedure:

  1. Starting near the top-most mark on your tape, measure and record the interval displacements between each spot (Dyi = yi – yi-1) in a table in Excel (or on paper). For example, measure the displacement between the top and point 1 (Dy1), then from point 1 to point 2, (Dy2), etc.

    Estimate the uncertainty in your displacement interval measurements, and calculate the % uncertainty of each relative to the interval measurement.

  2. Graph the instantaneous interval displacement versus elapsed time. Does the slope of this graph change? If it does, the object is accelerating!

  3. Calculate the instantaneous average velocity of the falling object at each point, and record in your table:

vi = Dyi/ t where t is 1/60th of a second

  1. Graph these instantaneous velocities of the falling block vs. time, and determine the slope of the curve. The value of the slope will approximate the value of “g”. (You can use Excel’s statistics functions to determine the value of the slope, as well as estimate its uncertainty statistically.)

    The percentage uncertainty in your estimate of the slope, “dgB” will again depend upon the uncertainty in your time measurements and your distance measurements. Since you took a number of distances to get the average instantaneous velocity, your average uncertainty in distance is divided by the square root of the number of data points you used (n).

    %dgA = (2 x % Uncertainty in one time measurement) + (Average % uncertainty in distance/√n)


Record both your value of the slope and its uncertainty as gB +/- dgB %.

 

 

 Experiment 2 Questions:

 

  1. What is the difference in results for “g” from Method A and Method B? Which is more precise? Which is more accurate? Why?

  2. Comparing your results from Experiment 1 and both versions of Experiment 2, which produced more precise estimates of “g”? Why? Which was most accurate? Why? What effect do you think human reaction time has in this experiment?


 


 

 

Chabot College Physics
Scott Hildreth

 

Free-Fall Tables for Experiment 2
Lab Data

 

 

 

 

 

Measurement Data

Calculated

Spot

#

Total Distance

from

“top” spot on

Tape

yi

Instantaneous

Displacement

Between Adjacent

Points

Dyi = yi – yi-1

Elapsed

Time

 

ti

 

Avg. Velocity

 

vi (avg) = 2yi/ ti

 

Instantaneous

Velocity

 

vi = Dyi/ t

 

 

cm

cm

sec

cm/sec

cm/sec

 

+/- cm

+/- cm

+/- sec

Displacement/
Time Interval

Distance/
Elapsed Time

 

 

 

 

 

 

1

 

 

 

 

 

2

 

 

 

 

 

3

 

 

 

 

 

4

 

 

 

 

 

5

 

 

 

 

 

6

 

 

 

 

 

7

 

 

 

 

 

8

 

 

 

 

 

9

 

 

 

 

 

10

 

 

 

 

 

11

 

 

 

 

 

12

 

 

 

 

 

13

 

 

 

 

 

14

 

 

 

 

 

15

 

 

 

 

 

16

 

 

 

 

 

17

 

 

 

 

 

18

 

 

 

 

 

19

 

 

 

 

 

20

 

 

 

 

 

21

 

 

 

 

 

22

 

 

 

 

 

23

 

 

 

 

 

24

 

 

 

 

 

25

 

 

 

 

 

26

 

 

 

 

 

27

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Chabot College Physics – Hildreth

Determining the Acceleration of Gravity

 

Experiment 3: Vernier Data Collection


Experiment 3 uses the automated data collection system created by Vernier Corporation, using LoggerPro software to analyze the data. I’ll walk you through how the system works, and how you can quickly generate velocity vs. time graphs for a falling object, analyze those graphs for the slope, and establish an experimental value for the acceleration of gravity.

In this experiment, you will have the advantage of using a very precise timer connected to the computer and a Photogate. The Photogate has a beam of infrared light that travels from one side to the other. It can detect whenever this beam is blocked. You will drop a piece of clear plastic with evenly spaced black bars on it, called a Picket Fence. As the Picket Fence passes through the Photogate, the computer will measure the time from the leading edge of one bar blocking the beam until the leading edge of the next bar blocks the beam. This timing continues as all eight bars pass through the Photogate. From these measured times, the program will calculate the velocities and accelerations for this motion and graphs will be plotted.



1. Observe the reading in the status bar of Logger Pro at the bottom of the screen. Block the Photogate with your hand; note that the Photogate is shown as blocked. Remove your hand and the display should change to unblocked.

2. Click to prepare the Photogate. Hold the top of the Picket Fence and drop it through the Photogate, releasing it from your grasp completely before it enters the Photogate. Be sure you are dropping the fence directly into the bucket below, cushioned with a towel to ensure the fence is not damaged.

Be careful when releasing the Picket Fence. It must not touch the sides of the Photogate as it falls and it needs to remain vertical. Click to end data collection.

3. Examine your graphs. The slope of a velocity vs. time graph is a measure of acceleration. If the velocity graph is approximately a straight line of constant slope, the acceleration is constant. If the acceleration of your Picket Fence appears constant, fit a straight line to your data. To do this, click on the velocity graph once to select it, then click to fit the line y = mx + b to the data.

Record the slope in the data table below.

4. To establish the reliability of your slope measurement, repeat Steps 2 and 3 at least five more times. Do not use drops in which the Picket Fence hits or misses the Photogate. Record the slope values in the data table.

Experiment 3 Data Table

 

Trial

1

2

3

4

5

6

Slope (m/s2)

 

 

 

 

 

 

 

 

Minimum

Maximum

Average

Acceleration (m/s2)

 

 

 

 

 

Acceleration due to gravity, g

m/s2

Precision

%

 

Experiment 3 Questions

g.      From your six trials, determine the minimum, maximum, and average values for the acceleration of the Picket Fence. Record them in the data table.

h.      Compare the precision and accuracy of this experiment with the first two. Overall which experiment produced the most precise data? Was that also the most accurate? Explain.