# Modeling Acceleration of a Two-component System

This lab is designed to align with AAOT science outcome #1: Gather, comprehend, and communicate scientific and technical information in order to explore ideas, models, and solutions and generate further questions.

## Materials

• writing utensil
• calculator
• digital device with spreadsheet program
• digital device with internet access

## Objectives

1. Apply Newton’s Second Law to model the acceleration a two-component system that is not in free-fall.
2. Analyze experimental data to test the acceleration model.
3. Recognize that the model assumes the equivalence of inertial and gravitational mass and decide if the experimental results provide evidence to support the mass equivalence.
4. Understand how Relativity Theory predicts the observed equivalence between gravitational and inertial mass.
5. Explain how the observation of a constant free-fall acceleration provides evidence for the equivalence of gravitational and inertial mass.

## Experimental METHODS

The following video shows a cart accelerated by a falling mass attached to the cart by a string running over a pully.  The experiment was repeated for 5 different cart masses, starting with the empty cart and successively adding four additional masses. The velocity of the cart was measured by the cart’s optical sensor. All of the necessary setup data is available in the video and the recorded velocity data is available in an online spreadsheet. You can copy and paste the data into your own sheet for analysis.

## Modeling

1) Assuming the track is level, what is the vertical acceleration of the cart?

2) In that case the net vertical force on the cart must be zero, so the force of gravity on the cart must be balanced by the  __________ force on the cart from the track.

3) We  also neglect friction and air resistance so the only horizontal force on the cart is the tension in the string. Write Newton’s Law for the cart in the horizontal direction using $F_T$ for tension and $m_c$ for the cart mass.

4)  Neglecting air resistance, write Newton’s Second Law for the hanging mass, $m_h$.

5)  This experiment is near the surface of Earth, so you may use $F_g = m_hg$ for the magnitude of the force of gravity on the hanging mass. Let’s choose downward as the positive direction for consistency.

6)  Your analysis of the cart gave you an equation for the tension. Assuming no friction in the pulley, the magnitude of the string tension must be the same on both ends. Substitute that result for the magnitude of the tension force in your current equation and set the sign as negative because tension points upward, which we chose as negative.

7) The magnitude of the acceleration for the hanging mass and cart are the same because they are connected by the string. Solve for the acceleration of the mass and cart.

8) We could have arrived at this same result by recognizing that the frictionless pully cannot change the tension in the string. Therefore, this situation is no different from pulling horizontally on the mass with the same force as its weight while it is connected directly to the cart with a horizontal string and no pulley. In that case the only external force on the system is the $F = m_hg$ and the system mass is $m_c +m_h$. Write Newton’s second law for this system and solve for acceleration to verify that we get the same result. Show your work.

9) According to our model, the acceleration should be proportional to $\frac{m_h}{m_h +m_c}$, with what proportionality constant?

## Data Analysis

First we will use the data in the online spreadsheet to find the acceleration for each trial. There are two methods we can use.

10) The first is to fit the plot velocity vs. time data for each trial and fit the data with a line. If the velocity vs. time data is linear then the ____________ of the fit line  will tell you the acceleration. If you choose this method be sure to plot and fit only a subset of each dataset beginning after the cart was released and ending before the hanging mass hit the ground.

11) The second is to find the maximum velocity reached for each trial and then use a kinematic equation to find the acceleration for each trial. You will need to know the distance travelled, which is shown at the end of the video. Show the work of symbolically solving the appropriate kinematic equation for the acceleration.

12) Choose one of the methods above and create a spreadsheet containing the trials and accelerations.

13) Make third column in your spreadsheet which calculates $\frac{m_h}{m_h +m_c}$ for each trial. The cart mass for each trial includes the total accessory mass used for that trial. Make sure these columns are well labeled, including units.

14) Plot the measured acceleration vs. $\frac{m_h}{m_h +m_c}$. Make sure the graph is named and the axes are labeled, including units.

15) Fit a trendline to the data. Record the fit equation and the r2 value here.

16) Does the fit line indicate that the relation between acceleration and $\frac{m_h}{m_h +m_c}$ is proportional as predicted by the model? Explain.

17) Calculate a % difference between the experimental slope and the expected slope (proportionality constant). Show your work.

## Conclusions

18) Do your experimental data support the acceleration model you created? Explain.

## Further Questions

19) What steps would you take to reduce uncertainty in the measurements for this lab? Explain exactly changes you would make or what equipment you would use, how, and why the uncertainty would be reduced (“I would do better” or “use better equipment” are not sufficient.)

20) Notice that this lab assumed that gravitational mass, used to calculate the gravitational force on objects, and inertial mass, used in Newton’s Second Law, were the same thing. The equivalence of gravitational mass and inertial mass leads to models which accurately predict what we observe. Consequences that have resulted from non-equivalence have not been observed. For example, in the previous lab you were able to model the range of a ball by assuming it’s free-fall acceleration was g, regardless of its mass. The equivalence of gravitational and inertial mass explains why all objects in free-fall near Earth will accelerate at the same rate of g regardless of mass. In this very lab you were able to model the acceleration of the system by inputting a value for inertial mass you found by measuring gravitational mass (with the scale).  Design your own experiment to test the equivalence of inertial mass and gravitational mass (other than a free-fall experiment). Describe/explain your experiment.

To learn about how Relativity Theory explains the equivalence of gravitational mass and intertial mass, watch this video.

To learn more and see why the equivalence principles do not suggest that Earth could be flat, watch the second video. To learn about the fundamental source of mass in the first place, watch the third and fourth videos.