# The Atwood Machine: Relationships Between the Masses and the Acceleration

The Rev. George Atwood (1746-1807) developed a simple structure known as the Atwood machine. It enables the researcher to test and verify the Newton’s Laws of Motion. This simple machine is made of two blocks connected by a light string running over a pulley that has no mass, which is suspended some distance above a floor or table (Mungan 15). If one of the masses is greater than the other (e.g., m1> m2), the system is pulled down by m1. With the assumption of a light, that is a pulley with no mass, and light strings that do not stretch, this system lends itself to analysis of Newton’s Laws. ## Newton’s Laws

Newton’s three Laws of Motion are significant to life. The main Law of focus in this study using the Atwood machine is the second Law (Goc 1). It notes that a body of mass m subjected to a force F and undergoes an acceleration a that has the same direction as the force and a magnitude that is directly proportional to the force and inversely proportional to the mass, that is, F = ma.

Hence, the acceleration of an object depends on the net applied force and the object’s mass. In an Atwood’s Machine, the difference in weight between the two hanging masses determines the net force acting on the system of the two masses. This resultant force accelerates both of the hanging masses; the heavier mass is accelerated downward, and the lighter mass is accelerated upward.

Thus mathematically;

F=d/dt(mv)

For bodies with constant mass;

F = m d/dt (v) = ma

## Purpose of the experiment

The purpose of the experiment is to determine the relationships between the moving masses mounted on the Atwood machine with the help of a Photogate.

## Results

### Graph Part I. ### Graph Part II ## Data Analysis

The difference between m1 and m2 was calculated and tabulated in the above tables for each part of the experiments this is shown by Δm. In addition, the total mass for both trials was calculated and tabulated as above and labeled mT.

The graphs from data obtained from both trials were plotted and are attached above. In the graph of acceleration vs. Δm, it was found that there is a linear relation between acceleration and change in mass. That means there is a linear proportionality between the two parameters. As the mass increases, so does the acceleration and this increases at the same rate. Hence, the mass difference is proportional to the acceleration of an Atwood machine.

In the second graph of acceleration vs. total mass, it was found that the acceleration increases exponentially with an increase in the total mass. This leads to the conclusion that acceleration of an Atwood machine is inversely proportional to the increase of total mass it bears.

Taking this below as a representation of the Atwood machine setup, the following expressions can be derived: T is the tension on the strings and inertia is assumed because the pulley is weightless.

∑F=ma

From the first trial, T-m1g= -m1a

From the second trial, T-m2g= m2a

T= m1g- m1a and T= m2a+ m2g

Therefore,

m1g – m1a= m2a- m2g

m1g – m2g= m1a+ m2a

(m1 – m2) g= (m1 +m2)a

a= (m1 m2) g/ mT

## Conclusion

From the results of the first experimental trial, it can be concluded that acceleration of the Atwood machine increases linearly with an increase in mass but having the total mass constant, that is the sum of m1 and m2 remain constant throughout the experiment with the indidual ones changing. And, in the second experimental trial, the acceleration of the Atwood machine increases exponentially with increase of total mass put on it. Some of these are that the pulley is massless and that there is no friction on the strings, which would have otherwise influenced the outcome of acceleration by reducing it.

## Works Cited

Goc, Roman. Atwood machine – testing Newton’s Laws of motion. Geostat: PL Publishers, 2005.

Mungan, Carl. Atwood’s Machine Without Hanging Masses. Annapolis: Annapolis Publishing Company, 2006.