The Atwood Machine                                                                                                                                                          

I.  Introduction

The Atwood machine is a device which was used by George Atwood to measure gravitational acceleration. This machine uses a pulley system and two unequal masses to test Newton’s second law of motion. Newton’s second law states that any system’s acceleration is directly proportional to the net external force which acts upon the system, and is inversely proportional to the total mass of the system.

 

                        ∑ F = ma

 

This experiment is also designed to measure the constant acceleration due to the force of gravity (g) acting on the machine.

 

II. Data Analysis

II.1 Measurement of Acceleration

 

In this experiment we measured the time (t) required for a system of two masses (m1 and m2) to fall a vertical distance (y). The difference between the two masses (m2-m1) was varied between 4 and 10 grams, while the sum (m1+m2) was kept at a constant 30 grams.

 

Sample Calculation for m2 = 20 grams and m1 = 10 grams:

 

*Average time t = 2.86 + 0.03 s,

 

where the uncertainty is the root mean square of the 4 independent time measurements.

 

*Acceleration a = 2 y / t2 = 42.8 m/s2

 

where the distance traveled is measured to be y = 175 cm.

 

*Uncertainty in the acceleration

 

            ∆a = 2 y / t2 – 2 y / (t + standard deviation)2

 

                 = 42.79 – 41.91 = 0.88 cm/s2

 

The measurement of the vertical distance y has a negligible uncertainty when compared to the uncertainty in t, the measurement of time. Because the uncertainty is relatively small, it is safe for us to neglect the effect of the y uncertainty on total uncertainty in the final measurement of acceleration due to gravity. The relative uncertainty in the measurement of time t :

 

                        ơt / t = 0.03 s / 2.86 s = 0.010 = 1.0%,

 

is much greater than the relative uncertainty in y,

 

                        ơt / y = 0.1 cm / 175 cm = 0.00057 = 0.06%

 

where the uncertainty in the distance measurement is conservatively estimated to be 1 mm.

 

II.2 Determination of Constant g

 

The graph of acceleration (a) as a function of the mass difference (m2-m1) is best depicted by the line of best fit, which shows a linear relationship between the two values. This linear relationship agrees with Newton’s second law:

 

            a (m1 + m2 + I / r2) = (m2-m1) g – f,

 

can be rewritten as

 

            a (m1 + m2 + M / 2) = (m2-m1) g – f,

 

where M corresponds to the mass of the pulley’s wheel.

 

The slope of the graph of a as a function of (m2-m1) is given by

 

            Slope = g / (m1 + m2 + M / 2)

 

From the graph, the slope of the best straight line through the data points is determined to be

 

            Slope = ∆y / ∆x = (0.64 m/s2 – 0.06 m/s2) / (14 grams – 3 grams) = 0.0527 m/(s2gram)

 

And thus, we can extract the value of g as follows

 

            g = Slope (m1 + m2 + M / 2) = 0.0527 m/ (s2gram) ∙ (30 grams + 332.2/2 grams)

                                                         = 10.34 m/s2,

 

where we used the known mass of the wheel M = 332.2 grams.

 

To calculate the uncertainty in our measurement of acceleration due to gravity, we can draw two straight lines through our data points. One line has a steeper slope than the best fit line, while remaining within the uncertainties of the data points (i.e. within the error bars). The slope of this steeper line is determined to be Slope+ = 0.0678 m/ (s2gram). Similarly, a line can also be drawn with a shallower slope than the best fit line. The slope of this shallow line is determined to be Slope- = 0.0364 m/ (s2gram). Both of these lines are shown as dashed lines on our graph, and yield the following values for acceleration due to gravity (g): g + = 13.29 m/s2 and g- = 7.14 m/s2. The uncertainty of these two values is one half of the difference, i.e. (g+ – g-) / 2 = (13.29 m/s2 – 7.14 m/s2) / 2 = 3.075 m/s2.

As a result, the acceleration of gravity is measure to be g = 10.34 + 3.075 m/s2.

 

II.3 Determination of Frictional Force f

 

The graph of our data points also allows us to calculate the friction force acting on the machine. The acceleration is a = 0 when the line of best fit intersects the x axis, and the relationship between the acceleration and the difference between mass can be simplified into the following expression:

 

            0 = (m2 – m1) g – f

 

which yields

 

            f = (m2 – m1) g­ = (1.8 grams) (10.34 m/s2)

                                    = 18.61 grams m/s2

                                    = 0.0186 N

 

 

III. Conclusion

 

Through the use of Atwood’s machine, we were able to test Newton’s second law. By measuring the acceleration of the masses tested with Atwood’s machine, we were able to calculate a value for the acceleration of gravity. While our data is not entirely linear, the line of best fit displays relatively accurate values for the force of gravity to which our machine was subjected to. We measured the acceleration of gravity to be 10.34 + 3.075 m/s2. The accepted value for the acceleration of gravity is 9.81 m/s2, which is within one standard deviation of our measured value. The data also allows us to calculate the frictional force acting on the system which is due to the rotation of the pulley around the axle; the value of this friction is 0.0186 N.