PACT Due Dates

Acceleration (Paul McCarty)

Title: Investigating the relationships of force, mass and acceleration

Principle(s) Investigated: 

  1. The relationship of the acceleration of a body and the magnitude of a force exerted on the body.
  2. The relationship of the acceleration of a body and the mass of the body.
  3. The relationships investigated above are integrated into Newton's Second Law of Motion

Standards : Past in the appropriate California content standards.
  • (1c) Students know how to apply the law F=ma to solve one-dimensional motion problems that involve constant forces (Newton’s second law).
  • (IEa) Select and use appropriate tools and technology (such as computer-linked probes, spreadsheets, and graphing calculators) to perform tests, collect data, analyze relationships, and display data.
  • (IEb) Identify and communicate sources of unavoidable experimental error.
  • (IEc) Identify possible reasons for inconsistent results, such as sources of error or uncontrolled conditions.
  • (IEd) Formulate explanations by using logic and evidence.
Materials
  • air track and air source (any number of science supply warehouses such as Flinn or Sargent-Welch)
  • gliders (same as above)
  • thread or very light test fishing line (sporting goods, fabric supply or general store such as target or walmart)
  • small washers or paper clips (hardware or office supply store)
  • pulley mounted to table or air track - the lower the mass the better (any number of science supply warehouses such as Flinn or Sargent-Welch)
  • stopwatch (science supply warehouse, sporting goods)
  • computers with internet connection or large white board to record student material
  • spreadsheet or hand calculators 


Procedure:

This Teaching Event consists of three parts. 1) First data is collected regarding the relationship of increasing force on a glider moving along an air track and its acceleration. 2) Then data is collected regarding the relationship of increasing mass of a glider moving along an air track and its acceleration. 3) Finally, this data is analyzed and synthesized to reveal the mathematical relationship, F=ma, Newton's Second Law.

Note: This procedure could be preformed any number of ways but I have chosen to minimize the amount of student manipulation of the apparatus, the air track. This is the first time the students have been introduced to the air track, but future investigations will allow for students to spend more time interacting with the track. Most teachers are faced with the limitations on the amount of money that can be spent on laboratory materials. Despite this challenge, I believe an investment in at least one track is still worthwhile as the air track is so effective at reducing the impact of frictional forces on the motion of the glider. Though students don't need to touch the air track in this investigation, they are still actively engaged in collecting and analyzing data.

Preparation:

An air-track is placed at the front of the class room high enough so that it is visible to all students in the room. It must be carefully prepared prior to the arrival of students. The end of the track must project over the end of a platform preferable 1.5 m high or higher. The track is leveled, a pulley is mounted at one end of the track directly in line with the motion of the glider moving down the track, and a thread is prepared by attaching one end to the glider, passing it over the pulley, and the connected the other end to a hanger for the masses.

It is also recommended that the instructor practice the procedure several times prior, adjusting the mass of the glider and the weight accelerating the glider until the thread does not visibly sag under its own mass and the acceleration of the glider is minimized to allow for the greatest possible time between the release of the glider and the end of the glider's acceleration.

The mass of the glider is determined. Masses are prepared to double, triple and quadruple the mass of the glider (make sure the glider still moves freely down the track at it's greatest mass. If the glider touches the track when fully loaded consider using other multiples of the original mass (instead of 1x, 2x, 3x, 4x consider 1x, 1.5x, 2x, 2.5x) to both lighten the maximum load while still allowing for at least four measurements for the Mass vs Acceleration portion of the lab.

The mass of the accelerating weight is determined and multiples of this weight are prepared and set aside to add to the hook at the end of the string. 

Student Procedure:

Divide the students into as many groups as is feasible. Each is supplied with a digital timer, a laptop computer to record materials, and their personal notebooks, calculators, and pencils.

After first demonstrating the procedure of collecting data and recording it followed by a couple practice rounds with students do the following:

Force vs. acceleration

  1. The teacher suspends the smallest accelerating mass at the end of the string and then at a count of three releases the glider.
  2. Students start timers when the glider is released and then stop the timers at some distance determined by their group but prior to the weight accelerating the glider hitting the floor or the end of the track, whichever comes first. (Encourage the groups to maximize the distance and time the glider travels to minimize the impact of errors in measurement.)
  3. Each group records the following information into a questionnaire directing the data to a spread sheet shown below.
    • Group Number: Each group is assigned a number to be able to hold them accountable for the quality of the data inputted into the questionnaire.
    • Force: recorded as 1F, 2F, 3F, 4F
    • Location (initial): every group should have the same initial location, the point at which the glider was released. Location is recorded in units of centimeters or meters.
    • Location (final): the final location of the glider will vary from group to group. Location is recorded in units of centimeters or meters.
    • Time: the time it takes the glider to travel from the initial location to the final location recorded above.
  4. The procedure is repeated with increasing weight until data is collected for all 4 forces.

Mass vs. acceleration

  1. The teacher prepares the glider with the lightest load and accelerated by the the lightest weight that has been determined sufficient to accelerate the glider once it is fully loaded.
  2. The glider is released on a count of three.
  3. Students start timers when the glider is released and then stop the timers at some distance determined by their group but prior to the weight accelerating the glider hitting the floor or the end of the track, whichever comes first. (Encourage the groups to maximize the distance and time the glider travels to minimize the impact of errors in measurement.)
  4. Each group records the following information into a questionnaire directing the data to a spread sheet shown below.
    • Group Number: Each group is assigned a number to be able to hold them accountable for the quality of the data inputted into the questionnaire.
    • Mass: recorded as 1m, 2m, 3m, 4m
    • Location (initial): every group should have the same initial location, the point at which the glider was released. Location is recorded in units of centimeters or meters.
    • Location (final): the final location of the glider will vary from group to group. Location is recorded in units of centimeters or meters.
    • Time: the time it takes the glider to travel from the initial location to the final location recorded above.
  5. The procedure is repeated with increasing mass until data is collected for all 4 masses of the glider.

Analysis

  1. Create a graph of mass vs acceleration and a graph of force vs acceleration.
  2. For both graphs determine whether there is a direct or inverse relationship between the variables.
  3. Combine the two proportions to create one proportion relating acceleration to force and mass then convert the proportion to an equation (a=F/m). Note: this is possible without the addition of a constant because the unit of force (N) is defined by the units of mass (kg) and acceleration (m/s2).
  4. Finally, test the equation. The teacher will provide the students with the mass of the system as well as the gravitational force being applied to the system. The students calculate the expected acceleration and then after a final run of the glider student predictions are compared against the experimental value of the acceleration.
  5. This final data collection along with the distribution of data collected earlier provides a wonderful opportunity to address sources of experimental error.


Student prior knowledge:

  1. Force is a push or pull on an object.
  2. When an object is accelerated a force is applied.
  3. A larger force is necessary to accelerate a more massive object at the same rate as a less massive object.
  4. Newton's equations describing the motion of an object undergoing uniform accelerated motion. Of particular importance is the equation distance = 1/2at2


Explanation

When Gallileo crafted his law of Falling Bodies he noted the confusion air resistance and other forms of friction produced in the minds of the both the casual observer and students of motion. It was due to these forces opposing motion that a long list of impressive students of motion dating as far back as Aristotle insisted that objects accelerated at an increasing rate as their mass increased. Frictional forces bring an additional level of complexity to motion. It is primarily for this reason an air-track is highly desirable for this investigation. Other apparatuses can be used such as dynamic cars and tracks but the air-tracks impressive reduction of frictional forces is a strong argument in its favor. The reduction of frictional forces also allows for lower speeds allowing for increased accuracy of in measurements of distance and time. 

Their are a number of ways to reduce experimental error beyond lower speeds. The greater the distance the glider travels the less impact errors in this measurement will have on the final numbers. Unavoidable sources of error include any warping of the track and glider, air resistance, and the rotational inertia of the pulley as well as friction in it's bearings. While I have not done any calculations to confirm this I doubt air resistance is even a measurable factor at the low speeds the track is operated at and with the use of hand operated timers. The inertial of the pulley can be reduced by reducing its mass while frictional forces can be reduced by lightly oiling the contact between the pulley axle and its support.

The acceleration of the glider is determined using the formula created by Newton which shows that distance can be described as a function of time. Using the calculus he created he found that the distance traveled by a uniformly accelerated object is the derivative of location with respect to time (s(t)=1/2at2), the velocity of the same object is the derivative of distance with respect to time (v(t)=at), and the acceleration of that same object is revealed to be independent of time when the derivative of velocity is calculated (a(t)=a). When the formula for distance (distance=1/2at2) is rearranged we can determine the acceleration when distance and time are known (a=2(dist)/t2). 

A bit of faith in their teacher is required by the thinking student who wonders at the source of the force for accelerating the object. How do you know that doubling the mass will simply double the force? After all, that is what we are trying to figure out. You can use the common experience we all have of object with greater mass exerting a greater force but that doesn't seem to answer either. Maybe someone with better deductive skills can answer this. I intend to do some research and see how Newton actually discovered his famous law.

Student data is automatically graphed to reveal the relationships of force and mass with acceleration. A direct relationship is revealed by a line with a positive slope. For instance, as force increases so does acceleration. An inverse relationship is revealed by a line with a negative slope. In the case of mass and acceleration, as the mass of an object increases the acceleration of the object by a constant force decreases. While the direct relationship above is linear, the graph of the inverse relationship will be a downward curve. If you were to graph the reciprocal of mass vs acceleration you would see a straight line with a positive slope.

The final formula is determined by combination of the two relationships

Discussion of error


Questions & Answers: Give three thought-provoking questions and provide detailed answers.

Applications to Everyday Life: Explain (don't just list) three instances where this principle can be used to explain other phenomenon.

  1. Rockets - acceleration increases as fuel burned due to the decreased mass of the rocket.
  2. constant acceleration of falling objects - Paul Hewitt's description of how increased mass leads to a greater gravitational attraction to the earth which should increase the acceleration but this increased force is countered with the increased inertia of a more massive object.
  3. The greater the mass of an object the greater its weight (force). 


Photographs: Include a photograph of you or students performing the experiment/demonstration, and a close-up, easy to interpret photograph of the activity --these can be included later.

Videos: Include links to videos posted on the web that relate to your activity. These can be videos you have made or ones others have made. 


https://docs.google.com/spreadsheet/viewform?formkey=dEJZQWpMakd1d0pDZERtWW8xdGdzOVE6MQ

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