**Materials**

1 each track, with 2 bumpers

1 each cart

1 each “smart pulley” with interface cable

2 each cart bar masses

thread and scissors

1 each small hanging mass sets

PC 850 interface

*Common Materials*: level (available from your lab instructor), electronic scale

**Goals**

1) To determine the acceleration of a mass when acted on by a net force using a computer and a Smart Pulley sensor for acquiring the data. Two cases are of interest: a) the mass is fixed and the net force is varied, and b) the force is fixed and the mass is varied.

2) To use graphical analysis techniques on the resulting data, and to check the validity of Newton’s second law of motion.

## Newton’s Second Law Introduction

Newton’s first law states that no change in the motion of an object can take place in the absence of a net force. In other words there is no acceleration (change in velocity) unless there is a net force. But how is the acceleration related to the force? Newton’s second law deals with this relationship. Experimentally we will explore the relationship between the net force on an object, the mass of the object, and the acceleration of the object due to the force. In this way we should be able to “discover” Newton’s second law of motion.

**Be sure to level the track carefully before you take any data!**

**Exercise 1: **

## Acceleration with a Varying Force and a Constant Mass

A small mass is connected to the cart by a string and hangs down over a pulley. If the “system” to be accelerated is considered to be the cart plus the mass hanging on the end of the string, then the only force causing this system to accelerate is the force of gravity on the hanging masses (ignoring any friction for the time being). This is possible because both the cart and the hanging mass do move together virtually the same as if they were glued together (assuming that the string is not stretchy).

With the use of a PASCO Smart Pulley (refer to the Smart Pulley section of the Computer Tools Supplement at the back of the lab manual for specific instructions on connecting it to the computer – or ask your lab instructor) the computer can take the appropriate timings and then compute distance, velocity, and acceleration as functions of time.

You may wish to plot all three of these quantities to determine the best method of determining the acceleration. Hint: the slopes of velocity-time graphs are constant when the acceleration is constant, and these slopes can easily be found along with an error estimate using the curve fitting tool in the menu at the top of the graph window.

The mass on the end of the string should be varied between 10 g and 60 g in 10 g increments. Since the combined mass of cart+hanger+weights is to be held constant, you will want to leave your unused 10g weights sitting on the cart, and move them to the hanger one a t a time. The mass hanger itself contributes 5 grams – don’t forget to add that in to your equations. *It is important that the hanging mass not exceed 60 g. Masses larger than 60 g can produce such large velocities that the equipment can be damaged.*To keep the mass of the system fixed make sure that any unused hanging masses ride in the cart!

Now make a graph of the acceleration as a function of the force. Describe the resulting relationship being as quantitative as you can. If possible, find a mathematical relationship (an equation, if you will) between the force and the acceleration of the cart. Your lab instructor can be of assistance here.

Exercise 2:

## Acceleration with a Constant Force and a Variable Mass

As in Exercise 1 we choose both the cart and hanging mass as the “system” to be accelerated. By keeping a fixed value of mass hanging on the string, the gravitational force causing the system to accelerate is kept constant. Use 50 or 60 g on the end of the string for this exercise (remember, the mass hanger is 5 grams). Heavier hanging weights, and additional rectangular steel bars (two maximum!) can be placed on top of the cart to increase the mass of the whole system.

Plot the acceleration as a function of the mass for at least 5 values mass. (Remember, to include the hanger and masses in the system mass still, even though they aren’t being varied here.) Qualitatively what happens to the acceleration when the mass increases?

Exercise 3:

### Putting It All Together

Newton’s second law states that the acceleration of an object with constant mass is directly proportional to the net force and that the acceleration of an object under a constant net force is inversely proportional to the mass. In mathematical form this looks like:

*a = F/m, *or, *F = ma*

If Newton’s law is correct, then you should be able to compute the mass of the “system” from your graphical analysis in Exercise 1. Ask your lab instructor for assistance if it is not clear how to proceed. Compare your calculated mass with the measured total mass of the “system.”

The relationship between the acceleration and the mass in Exercise 2 is a bit trickier. Graphs that are not straight lines are rather difficult to analyze since the mathematical connection between the variables is hard to guess from looking at the shape of the graphical relationship. On the other hand straight-line graphs are simple to identify and analyze.

Is there a way of plotting the acceleration and mass in Exercise 2 consistent with Newton’s second law such that the graph will be linear? Consult the “Uncertainty and Graphical Analysis Supplement” for hints. When you have sorted this out, you will be able to compute the value of the force applied to the system, form your graph. Compare this value to the gravitational force on the hanging mass.

#### Newton’s Second Law Conclusion

Was Newton right? Are there reasons that our results might not be totally consistent with the predictions of Newton? Have we included all the forces acting on each system that we have analyzed today? What effect does the inevitable frictional force have? In looking back at your graphs of force and acceleration and of acceleration and mass, can you see where the presence of other forces is apparent? Explain.

Here’s a question to think about. How was Newton able to formulate the second law of motion? Did he have access to equipment that was superior to what you used today so that he was compelled to formulate the law based on the experimental results? Hmmm….

For the benefit of people that have lab after you, before you leave the lab:
Quit all computer applications that you may have open. Gather together the set of small masses to hang on the string. Report any problems or suggest improvements to your lab instructor. |

Ex-1 | g | cart g | a | uncertainties | cart- 256.9g | ||

d | 10g | 60g | 0.364 | 0.0011 | |||

20g | 50g | 0.656 | 0.0014 | ||||

30g | 40g | 0.936 | 0.0024 | ||||

40g | 30g | 1.22 | 0.0043 | ||||

50g | 20g | 1.51 | 0.005 | ||||

60g | 10g | 1.78 | 0.0066 | ||||

cart- 256.9g | |||||||

Ex-2 | Mh | cart g | a | uncertainties | |||

50g | 0g | 1.6 | 0.0033 | ||||

50g | 1.38 | 0.0049 | |||||

100g | 1.22 | 0.0034 | |||||

150g | 1.09 | 0.0033 | |||||

200g | 0.975 | 0.0026 |

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