Analysis of the Atwood Machine with Unequal Masses: Tension and Acceleration

Introduction to the Atwood Machine

The Atwood machine is a classic apparatus in classical mechanics used to demonstrate acceleration due to gravity. It involves two masses hanging from either side of a pulley, connected by a string. By analyzing the forces acting on these masses, we can calculate the tension in the string and the acceleration of the system. In this article, we will focus on a specific scenario where one mass is one-third of the other mass.

Deriving Tension and Acceleration

Let's consider the Atwood machine with two masses, ( M_1 ) and ( M_2 ), where ( M_1 frac{M_2}{3} ). We will use Newton's second law to analyze the system.

Step 1: Setting Up the Equations

First, we need to set up the equations for the forces acting on the masses. Let's denote:

( M_1 frac{M_2}{3} ) ( M_2 ) is the other mass ( g ) is the acceleration due to gravity

According to Newton's second law, the forces acting on each mass can be expressed as:

For ( M_1 ) (moving upwards): ( T - M_1 g M_1 a ) For ( M_2 ) (moving downwards): ( M_2 g - T M_2 a )

Step 2: Substituting ( M_1 )

Substitute ( M_1 frac{M_2}{3} ) into the first equation:

[ T - frac{M_2}{3} g frac{M_2}{3} a ]

Rearranging this equation gives:

[ T frac{M_2}{3} g frac{M_2}{3} a quad text{(1)} ]

Step 3: Combining the Equations

Now, substitute equation (1) into the second equation:

[ M_2 g - left( frac{M_2}{3} g frac{M_2}{3} a right) M_2 a ]

Simplifying this:

[ M_2 g - frac{M_2}{3} g - frac{M_2}{3} a M_2 a ]

Combining the terms:

[ frac{3M_2}{3} g - frac{M_2}{3} g M_2 a frac{M_2}{3} a ]

Factoring out ( M_2 ) on the right-hand side:

[ frac{2M_2}{3} g M_2 left( a frac{a}{3} right) ]

Simplifying the equation further:

[ frac{2M_2}{3} g M_2 left( frac{4a}{3} right) ]

Step 4: Solving for Acceleration

Dividing both sides by ( M_2 ) (assuming ( M_2 eq 0 )):

[ frac{2}{3} g frac{4a}{3} ]

Multiplying both sides by ( frac{3}{4} ) to solve for ( a ):

[ a frac{2g}{4} frac{g}{2} ]

Step 5: Substituting ( a ) Back to Find Tension ( T )

Substitute ( a frac{g}{2} ) back into equation (1):

[ T frac{M_2}{3} g frac{M_2}{3} left( frac{g}{2} right) ]

Combining the terms:

[ T frac{3M_2 g}{6} - frac{M_2 g}{6} frac{2M_2 g}{6} frac{M_2 g}{3} cdot frac{2}{2} frac{M_2 g}{2} ]

Thus, the acceleration ( a ) and tension ( T ) in terms of ( M_2 ) and ( g ) are:

Acceleration: ( a frac{g}{2} ) Tension: ( T frac{M_2 g}{2} )

Conclusion

In this analysis, we derived the tension and acceleration in the string of an Atwood machine where one mass is one-third of the other. This detailed step-by-step approach can be applied to more complex scenarios involving pulley friction or pulley inertia. Essentially, the same principles of Newton's second law and free body diagrams are used to solve for these values.

For further reading, you may explore the following keywords:

Atwood machine Tension Acceleration