Newton's Second Law

Objective: The objective of this lab is to confirm Newton's Second Law, f = m*a. Two activites are conducted. The first collects acceleration while the force creating the acceleration is held constant. The second measures acceleration while the mass of the system is held constant.

Equipment: Pasco cart track and cart Smart Pulley Photogate Mass hanger and masses TI-83 series calculator CBL or LabPro Scales to mass cart

Setup:

The pasco cart, track, smart pulley and photogate are setup as shown in the images above. The photogate is attached to a Vernier LabPro or Texas Instruments CBL. The LabPro or CBL is connected to a TI-83 series calculator which runs a SMPULACC program which also requires a LINKCK program. This program is available by contacting jwelker@lps.org. Please specify whether you are using the CBL or LabPro and whether you are using Macintosh or Windows platform for TI Graph-Link software. The SMPULACC program collects distance vs time data as the string crosses the smartpulley. It assumes students can use a calculator to find the quadratic regression and interpret the results.

Program Use Instructions for Data Collection:

1. Send the programs SMPULACC and LINKCK to the TI-83 Calculator.
2. Connect the photogate and picket fence as shown above to the CBL or LabPro
3. Run the program SMPULACC
4. Enter the number of revolutions the wheel will make during data collection. Press ENTER.
5. Ready the experiment. The calculator will display a message to press ENTER to arm the photogate. Press ENTER, release the cart allowing it to accelerate down the ramp.
6. If using the LabPro, you must press ENTER following data collection to get the data from the LabPro and the graph will be displayed. The CBL will automatically get the data and display the graph. Check the graph to make sure the data collection process was accurate. Moving the cart triggers the data collection. If the cart is moved, held for a short time and then released, the data may be inaccurate. In this case, collect another data sample.
7. Press ENTER to calculate the quadratic regression and find the acceleration. The acceleration is two times the value of a in the quadratric regression ax^2 + bx + c.
8. Press ENTER to run another trial.

Activity 1: Constant Force

1. Load the cart with a series of masses. Place enough mass on the hanger over the smart pulley to accelerate the system. Find the mass of the entire system (cart, masses on the cart, and mass of the hanger).
2. Run 10 trials removing mass from the cart on each trial. Instructions for determining acceleration are noted above. Record the mass of the system and the acceleration along with the mass on the hanger creating the force to accelerate the cart. The mass on the hanger should remain constant throughout the trials so the force remains constant.
3. Graph acceleration (y-axis) as a function of the mass (x-axis) of the system.
4. Find a model for the data set.
5. Explain the mathematical model.

Activity 2: Constant Mass

1. Load the cart with a series of masses. Place enough mass on the hanger over the smart pulley to accelerate the system. Find the mass of the entire system (cart, masses on the cart, and mass of the hanger).
2. Run 10 trials removing mass from the cart and placing it on the hanger for each trial. In this manner, the mass of the system will remain constant but the force will be increasing. Instructions for determining acceleration are noted above. Record the acceleration, mass on the hanger creating the force to accelerate the cart, and the mass of the system.
3. Graph acceleration (y-axis) as a function of the force (x-axis) applied by the mass hanger. (Note: The mass (kg) of the hanger must be converted to force (N).)
4. Find a model for the data set.
5. Explain the mathematical model.

Analyzing Constant Force - Acceleration vs Mass of System:

The data shown below is for the mass of the system and the acceleration of the cart. The hanger was a constant 0.05 kg. The graph of the accleration as a function of the mass of the system is also shown below.

Mass (kg) Accel (m/s^2) 1.002 0.402 0.952 0.428 0.902 0.455 0.852 0.476 0.802 0.511 0.752 0.562 0.702 0.592 0.652 0.636 0.602 0.693 0.552 0.775

The graph appears to be an inverse (reciprocal) relationship. To linearize the graph, we will graph the acceleration as a function of 1/Mass. This graph is shown at the right. The linearized regression equation is y = 0.45x - 0.05. The units of the slope are: The units of the slope are Newtons, hence the model is: Acc (m/s^2) = 0.448 (N) * 1/M (kg) - 0.045 (m/s^2)

The slope of the graph is constant and the units of slope are Newtons or force. The force was constant in the lab. The hanger contained 0.050 kg or 0.49 N. The difference between the constant force and the slope is 8.5%. With a small mass on the hanger, there may be significant friction to overcome in starting the system which helps to explain the difference between the slope (or calculated force) and the force on the hanger.

Fitting the model with an inverse function, one finds the same model as the linear regression fits the data. The model is: Acc (m/s^2) = 0.448 (N) * 1/M (kg) - 0.045 (m/s^2)

The y-intercept would be the acceleration of the system when the mass is zero. This would be expected to be zero. The % error is negligible in this case. With the slope of the line being force, the model suggests:

Analyzing Constant Mass - Acceleration vs Force:

The data set shown below resulted from trials beginning with 2.0031 kg for the mass of the system starting with a mass of 0.05 kg on the hanger creating the acceleration of the cart for the first trial. Mass was removed from the cart and placed on the hanger for each sucessive trial. Force was determined by mutiplying the mass in kilograms on the hanger by 9.8 m/sec^2 to get Newtons for the units of force.

Force (N) Accel (m/s^2) 0.98 0.429 1.96 0.911 2.94 1.402 3.92 1.889 4.90 2.290 5.88 2.801 6.86 3.287 7.84 3.776 8.82 4.218 9.80 4.682

The graph of acceleration as a function of mass of the system is as shown. The model appears to be linear with a regression of y = 0.48x- 0.03. Adding units to the model would result in Acc _(m/s^2) = 0.48 _(N/(m/s^2)) * F _(N) - 0.03 _m/s^2.

The units of the slope simplify to:

The model simplifies to Acc _(m/s^2) = 0.48 _(1/kg) * F _(N) - 0.03 _(m/s^2). The y-intercept would be expected to zero because a zero force applied to the hanger would be expected to result in no acceleration. The % error in the y-interecept is exceptionally small (0.7%) when compared to the largest y-value. The model is now simplified to Acc _(m/s^2) = 0.48 _(1/kg) * F _(N). The slope of the line is a constant and the units are 1/kg. As the force increased, the acceleration increased at a constant level. This constant is the reciprocal of the mass. The mass of the system was 2.0031 kg. The reciprocal of this mass is 1 / 2.0031 kg or 0.4992 (1/kg). The resulting error is 3.4% which may result from friction on the track and the smart pulley as well as the fact that each mass was not measured precisely on the hanger. The model implies: