Title: Investigating the relationships of force, mass and acceleration
Standards : Past in the appropriate California content standards.
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.
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.
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:
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
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