Measuring g, Exploring Free Fall
Inquiry Lab Kit for AP® Physics 1
Materials Included In Kit
Foam sheets, 8½" x 5½", 15 Spheres, steel, 1-5/16" diameter, 12
Spheres, steel, ⅝" diameter, 12
Additional Materials Required
(for each lab group) Balance, 0.01-g precision (may be shared) Basket or box Clamp holder Masking tape
Meter stick Photogate timer Picket fence Support stand Timer, stopwatch
Safety Precautions
The materials in this lab are considered nonhazardous. Follow all laboratory safety guidelines.
Disposal
All materials may be saved and stored for future use.
Lab Hints
- This laboratory activity can be completed in two 50-minute class periods. It is important to allow time between the Introductory Activity and the Guided-Inquiry Activity for students to discuss and design the guided-inquiry procedures. Also, all student-designed procedures must be approved for safety before students are allowed to implement them in the lab. Prelab Questions may be completed before lab begins the first day.
- The Introductory Activity may be presented as a teacher demonstration or conducted as a group activity. This investigation has been written to be educational and economical. If multiple photogate timers are available, the teacher demonstration/group activity may be run as an introductory experiment by groups of 3–4 students. Moreover, the introductory activity can be converted to a guided-inquiry activity by omitting procedural information. The level of difficulty can be increased by withholding information or decreased by providing hints.
- Should photogate timers be unavailable, the raw data provided herein may be used to facilitate calculations and discussions concerning uncertainty in measurement. The following web address leads to a video of a picket fence being dropped through a photogate timer. If no photogate timers are available, the video may be useful to students as a visual representation of the experimental process: http://www.flinnsci.com/teacher-resources/all-videos/all-videos/psworks-photogate-timer,-ap6998/ (accessed May 2014).
- This investigation is meant to introduce students to the kinematic equations describing one-dimensional motion. In addition, the investigation serves as a topical introduction to experimental error (random and systematic) and the empirical analysis of experimental error. For example, students should come to view standard deviation as a means to detect random error and quantify an experiment’s precision. Students should come to view percent error as a means to quantify accuracy. The investigation introduces students to mathematical manipulation of the kinematic equations and to experimental design problems (e.g., the difficulty associated with measuring free-fall times manually).
- The photogate timer will store 20 “readings” in its memory. In “one gate” mode the photogate timer will store data for five drops of the picket fence because each drop results in four distinct beam interruptions and time readings. The “memory” button can be used to scroll through the time data and the “reset” and “memory” buttons may be pressed simultaneously to clear the memory so that further data may be collected.
Teacher Tips
- To calculate g with photogate data obtained in one-gate mode, use the following equation:
g =(s/2n)(1/tf2 – 1/ti2)
where s = the width of one black and clear band interval of the picket fence, tf = time interval through the last picket (or fourth time displayed in one gate mode), ti = time interval through the first picket (or first time displayed in one-gate mode), n = number of complete black and clear intervals (black and clear band = 1) between the total distance measured. The picket fence-based calculations reported herein use the following values: s = 4.97 cm and n = 6. The tf and ti values vary and are reported in tabular form in the Sample Data and Discussion section for the Introductory Activity.
Further Extensions
Opportunities for Inquiry
Drop objects from high locations such as gymnasium bleachers or accessible rooftops. Measure the effects of air resistance on free-fall times using coffee filters. Similarly, add a loop of clear plastic tape to the top of a picket fence to create drag. Gently “throw” a picket fence or sphere up or down prior to its passing through a photogate to determine how such an act affects g.
Alignment to the Curriculum Framework for AP® Physics 1
Enduring Understandings and Essential Knowledge A gravitational field is caused by an object with mass. (2B) 2B1: A gravitational field g at the location of an object with mass m causes a gravitational force of magnitude mg to be exerted on the object in the direction of the field.
All forces share certain common characteristics when considered by observers in inertial reference frames. (3A) 3A1: An observer in a particular reference frame can describe the motion of an object using such quantities as position, displacement, distance, velocity, speed, and acceleration.
Classically, the acceleration of an object interacting with other objects can be predicted by using a = ΣF/m. (3B) 3B2: Free-body diagrams are useful tools for visualizing forces being exerted on a single object and writing the equations that represent a physical situation.
Learning Objectives 2B1.1: The student is able to apply F = mg to calculate the gravitational force on an object with mass m in a gravitational field of strength g in the context of the effects of a net force on objects and systems. 3A1.1: The student is able to express the motion of an object using narrative, mathematical, and graphical representations. 3A1.2: The student is able to design an experimental investigation of the motion of an object. 3A1.3: The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. 3B2.1: The student is able to create and use free-body diagrams to analyze physical situations to solve problems with motion qualitatively and quantitatively.
Science Practices 1.4 The student can use representations and models to analyze situations or solve problems qualitatively and quantitatively. 1.5 The student can reexpress key elements of natural phenomena across multiple representations in the domain. 2.1 The student can justify the selection of a mathematical routine to solve problems. 2.2 The student can apply mathematical routines to quantities that describe natural phenomena. 3.1 The student can pose scientific questions. 4.1 The student can justify the selection of the kind of data needed to answer a particular scientific question. 4.2 The student can design a plan for collecting data to answer a particular scientific question. 4.3 The student can collect data to answer a particular scientific question. 4.4 The student can evaluate sources of data to answer a particular scientific question. 5.1 The student can analyze data to identify patterns or relationships. 5.2 The student can refine observations and measurements based on data analysis. 5.3 The student can evaluate the evidence provided by data sets in relation to a particular scientific question. 6.1 The student can justify claims with evidence. 6.2 The student can construct explanations of phenomena based on evidence produced through scientific practices. 6.4 The student can make claims and predictions about natural phenomena based on scientific theories and models.
Correlation to Next Generation Science Standards (NGSS)†
Science & Engineering Practices
Asking questions and defining problems Developing and using models Planning and carrying out investigations Analyzing and interpreting data Using mathematics and computational thinking Engaging in argument from evidence Obtaining, evaluation, and communicating information
Disciplinary Core Ideas
HS-PS2.A: Forces and Motion HS-ETS1.B: Developing Possible Solutions
Crosscutting Concepts
Cause and effect Patterns Scale, proportion, and quantity Systems and system models
Performance Expectations
HS-ETS1-4. Use a computer simulation to model the impact of proposed solutions to a complex real-world problem with numerous criteria and constraints on interactions within and between systems relevant to the problem. HS-PS1-2. Construct and revise an explanation for the outcome of a simple chemical reaction based on the outermost electron states of atoms, trends in the periodic table, and knowledge of the patterns of chemical properties. HS-PS2-2. Use mathematical representations to support the claim that the total momentum of a system of objects is conserved when there is no net force on the system. HS-PS2-4. Use mathematical representations of Newton’s Law of Gravitation and Coulomb’s Law to describe and predict the gravitational and electrostatic forces between objects.
Answers to Prelab Questions
- If a feather and a hammer are dropped at the same time on Earth, which will hit the ground first? What if they were dropped on the Moon? Mars? Justify your answer.
On Earth and Mars, the hammer will hit first as they both have an atmosphere that will create air resistance that slows the feather more than the hammer. On the moon, they will both hit the ground at the same time since the moon has no atmosphere.
- If the acceleration due to gravity on the moon is 1.6 m/s2, how long will it take a feather to fall 1 m? What about the hammer?
a = 2d / t2 t = (2d / a) ½ t = [(2 x 1 m) / 1.6 m / s2]½ t = 1.1 s
- Describe potential sources of error in determining g using a photogate timer and picket fence? What might be done to minimize these?
Systematic error attributable to an instrument flaw tends toward being too high or too low and would be apparent if readings were intuitively wrong or if one instrument consistently gave significantly and statistically different readings than the other instruments. For example, if one photogate reported illogically high times (such as in the ten-second range), such a circumstance would alert the user to an instrument flaw. In addition, if five student groups reported readings in the 0.0198 s to 0.0598 s range, and one student group reported readings in the 2.345 s to 6.357 s range, such a circumstance would alert the user to a systematic flaw. The only way to correct for a systematic error is by applying a correction factor or recalibrating the faulty instrument.
In addition, random error may arise from tilting the picket fence. This results in a distance measured by the photogate beam that is larger than the perpendicular distance assumed in computations. This may be mitigated by dropping from some sort of makeshift mechanical mechanism such as a suspended vice.
Another source of random error is the unintentional imparting of some initial, albeit minor, velocity to the picket fence. Again, this type of error may be eliminated by dropping from a mechanical mechanism suspended above the photogate on a ring stand or other support.
- Describe potential sources of error in determining g using steel spheres and a manually controlled timer? What might be done to minimize these?
Significant random error is attributable to reaction time. This is compounded by the fact that the timekeeper does not know the exact moment at which the dropper will release the sphere. The best way to minimize random error is to increase the number of measurements taken. If the drop distance is increased, frictional forces due to air resistance also increase. If the drop distance is decreased, timing errors become more significant. It is best to perform a large number of trials so that random error is averaged to a reasonable mean and may be detected via standard deviation calculations.
- An archer fires ten arrows at a circular target. One of the arrows comes close to the bull’s eye but does not hit it. The other nine arrows contact the target in close proximity to each other but away from the bull’s eye. Describe the accuracy and precision associated with the archer’s shots.
The single shot that hit the bull’s eye may be called accurate. However, the average location of the shots is inaccurate because the majority of the shots land far from the bull’s eye, or true value. This “experiment” is fairly precise as nine of the ten shots cluster near each other and the archer shows a high degree of reproducibility. A situation in which all ten shots strike the bull’s eye or land in close proximity to the bull’s eye and to each other would be called accurate and precise. Precision is a measure of the closeness of a data set’s values to each other.
- The following are experimental measurements of water’s density, obtained by massing a measured volume of water: 1.000 g/ml, 1.220 g/ml, 1.200 g/ml, 0.980 g/ml, 1.080 g/ml and 1.150 g/ml. Calculate the standard deviation and percent error for this data set and use the calculated values to assess the experiment’s accuracy and precision. Assume the density of water is 1.00 g/mL.
{12481_Answers_Table_1}
Based on the fairly low standard deviation value of 0.1016, it can be said that the experiment was conducted with a fairly high degree of precision. There is no specific, universal standard deviation value correlated to very high precision or very low precision. Rather, assessments of precision are relative. For example, if six other student groups carried out the density experiment described above and calculated an average standard deviation of 0.005, the value 0.1016 would imply a lower level of precision, but not necessarily a “low” level of precision.
The percent error was calculated by dividing the absolute value of the difference between the experimental density and accepted density by the accepted density and subsequently multiplying the quotient by 100: [(1.105 – 1.00)/1.00] x 100 = 10.5%. Given this fairly high percent error value, it can be said that the average experimental value is not very accurate. That is, it is significantly different than the known density of water.
Sample Data
Equation 5 is used to calculate the acceleration due to gravity when using the picket fence and photogate in one-gate mode. The calculations are based on the following values: s = 0.0497 m and n = 6.
{12481_Data_Table_2}
For Trial 1
g = (s/2n)(1/tf2 – 1/ti2 ) g = (0.0497 m/2 x 6)[(1/0.0193 s)2 – (1/0.0553 s)2] g = 9.76 m/s2 % Error = |Experimental – Theoretical|/Theoretical x 100 % Error = |9.76 m/s2 – 9.81 m/s2|/9.81 m/s2 x 100 % Error = 0.463%
To calculate standard deviation: subtract each g value from the mean (9.767 m/s 2), square each of the differences, add the squared differences, divide the sum by n – 1, and take the square root of the quotient:
{12481_Data_Equation_7}
Guided-Inquiry Design and Procedure
{12481_Data_Table_3}
{12481_Data_Table_4}
Accuracy and Precision When the standard deviation and percent error values from the picket fence drop and sphere drop are compared, the picket fence drop showed a much greater degree of accuracy and precision. The sphere drop had significant variability in the value of g between trials, indicated by a larger value for the standard deviation. A comparison of the standard deviations indicates that the sphere drop was less precise. Additionally, the percent error values for the sphere drop were 11.2% and 6.5%. The picket fence drop had a percent error of ≤1%. The sphere drop was less accurate than the picket fence drop.
Answers to Questions
Guided-Inquiry
- Based on the calculated standard deviation and percent error, assess the accuracy precision of the experiment in the Introductory Activity.
The standard deviation for the Introductory Activity was 0.0654. This is a low value for a standard deviation and indicates a high degree of precision. The 10 experimental values of g are close together in value.
The percent error for each trial is shown in the data table on the next page. The majority of the ten trials are close to the accepted value (9.81 m/s2). All trials have percent errors of 1% or less. This indicates the experiment was highly accurate.
- According to the Background, Galileo asserted, and was later proved correct, that objects of differing masses fall at the same rate. Using a free-body diagram, explain why two spheres of mass m and 2m will fall at the same rate.
{12481_Answers_Figure_1}
There is only force acting on the spheres: the force due to gravity, Fg. The acceleration on both spheres is g, 9.81 m/s2, and causes the spheres to fall at the same rate.
- When using a stopwatch, there is a small amount of time between observing an event and pressing the start/stop button.
Describe how random errors in timing could be minimized.
In order to minimize random errors in timing, multiple “timers” can be used. If the two times differ greatly, then the trial should be thrown out. Additionally, a specific countdown method can be used to sync the “timers” and sphere “dropper,” such as “3...2...1...drop.” The “timers” start on “drop” and stop when the sphere hits the ground. Another method to be sure the “timers” are in sync is conduct practice trials. To mimic the actual experiment, the countdown method can be used to start the “timers.” Then the “dropper” can tap the steel sphere on the lab table or ground to signal the end of the trial. The times are compared to gauge how well the “timers” hear the sphere hitting the ground. It may also be advantageous for the “timers” to keep their eyes closed so they do not prematurely stop the timers by predicting when the sphere hits the ground.
A single trial is not enough to guarantee accurate fall times. At least 10 trials should be conducted with little variability between the times. When the fall times are in agreement, it can be assumed that the acceleration due to gravity calculated using those times will be accurate and precise.
- Does one trial provide sufficient data to determine the acceleration due to gravity to a reasonable degree of accuracy and precision? Explain your reasoning.
A single experimental value may be compared to the accepted value of g and the accuracy of the single experiment assessed. However, the precision and reproducibility of the experimental design cannot be assessed with a single trial. At least two trials are needed to assess precision. As the number of trials increases, an assessment of precision becomes better.
- Consider the experiment conducted by Galileo in the Background section. Design an experiment to test Galileo’s findings. Identify the independent and dependent variables, any variables that should remain constant, and other factors that may have an effect on the reproducibility of the experiment.
Two spheres of differing mass are dropped from the same height by using a meter stick. The fall times are measured using a stopwatch. The spheres are massed on a balance and values recorded. One sphere is held above a “catcher’s” box that contains foam squares to protect the floor. The height of the sphere is recorded. The timers are readied and primed by using the “3...2...1...drop” countdown. Timers start on “drop” and stop when the sphere hits the ground. Times are compared and recorded. If the times vary greatly, the trial is thrown out. A total of 10 trials with agreement between timers are conducted. The procedure is repeated for the other sphere.
Review Questions for AP® Physics 1
- Galileo carried out his seminal free-fall experiment using an inclined plane. Why is such an approach feasible, and sometimes preferable, to the simple drop method for determining g?
Measuring g using an inclined plane may be preferable because it extends the object’s free fall time, particularly if the plane has a very shallow slope, and reduces the impact of random error on the calculation of g. It is scientifically reasonable to carry out the experiment in this manner because a sphere released from rest on an inclined plane has initial velocity equal to zero and the force of friction which acts on the sphere as it rolls is negligible, particularly when compared to the friction generated by air resistance. Since gravity is the only significant force that acts on the sphere, its acceleration due to gravity may be calculated by measuring free-fall time.
- Would “throwing” the picket fence up or down prior to its passing through the photogate affect the experimentally determined g value?
Throwing the picket fence down, or imparting a downward velocity prior to its passing through the photogate would lead to high experimentally determined g values. In contrast, throwing the picket fence up and then allowing it to fall back down through the picket fence would not affect the experimentally determined g values because the picket fence would have an initial velocity (in the downward direction) equal to zero.
- A student performed a sphere drop experiment. The sphere was dropped from 1.50 m above the ground and the sphere has a mass of 0.875 kg. Free fall times for the student’s experiments follow.
- Before the student calculates the acceleration due to gravity from the data, are there any trials that should be thrown out? Justify your choice.
The second trial, t = 0.700 s, should be thrown out and not used to calculate the acceleration due to gravity. The theoretical time for the sphere to fall a distance of 1.50 m is 0.553 s. The second trial does not align with the theoretical time. The other four trials are in closer agreement in time and show a high degree of reproducibility. The second trial is a random error.
- Calculate the average acceleration due to gravity from the student’s data.
{12481_Answers_Table_5}
Average time = 0.554 s a = 2d/t2 a = 2 x 1.50 m/(0.554 s)2 = 9.77 m/s2
- Which of the following graphs represent how the velocity of the sphere changes over time when falling with constant acceleration? Justify your choice.
{12481_Answers_Figure_2}
Graph B depicts the correct change in velocity over time for a sphere falling with constant acceleration. As the sphere accelerates, each passing second causes the sphere’s velocity to change by a x t.
Graph A shows that the velocity of the sphere is constant. If something is moving with a constant velocity, then it is not accelerating. However, the falling sphere is accelerating so velocity should be changing.
Graph C shows velocity is increasing but not in a linear manner. Velocity is equal to acceleration multiplied by time. The graph appears to be a squared function. This means the acceleration is changing over time.
Graph D shows that velocity changes at some time. The sphere is accelerating up until the peak of the graph. At the peak, the sphere begins to have a negative acceleration and slows down as velocity goes back to 0 m/s. The acceleration is not constant if it changes direction.
References
AP® Physics 1: Algebra-Based and Physics 2: Algebra-Based Curriculum Framework; The College Board: New York, NY, 2014.
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