Method to Determine the Acceleration of Gravity is to Use an Atwood's Machine

Atwood’s Machine INTRODUCTION One method to determine the acceleration of gravity is to use an Atwood's machine, a device that consists of two masses connected by a string and suspended over a pulley as illustrated in the figure below. Interpretation of the results will emphasize the usefulness of the graphical analysis of data. THEORY Atwood’s machine consists of two masses (

and

) connected by a light string that passes over a pulley of mass

and radius
, as shown in the Figure 3.1. The average friction f due to the bearings is exerted on the pulley’s axle which has a radius



. We use Newton’s 2nd law to determine the acceleration
of each mass. Because no forces act on either

or

in the horizontal direction (

direction), we only need to apply Newton’s 2nd law in the vertical direction (

direction). For

we get















































































so that Figure 3.1: Atwood's machine setup
















































































where
is the acceleration due to gravity, and T1 and T2 are the tensions on

and

, respectively. Applying Newton’s 2nd law to m2 yields















































































giving
















































































If friction and the mass of the pulley are negligible, then






, and it is not difficult to solve Eqns. (2) and (4) for
. The result is



















































































In practice, it is not easy to design an experiment where friction and the pulley’s mass are negligible, so we need to develop an expression that shows the effects of force f and

. This may be done by introducing Newton’s 2nd law for rotating objects given by












































































where
is the torque,
is the rotational inertia, and
is the angular acceleration, which is related to the linear acceleration by



, where
is the radius of the rotating object. For the case under consideration here, torque is simply related to the force and the radius by


. Using this definition of torque, the equation governing the motion of the pulley is given by












































































If we combine Eqns. (2), (4), and (7),






, and



, we obtain an expression for
given by







































































where

is the effective mass of the pulley, taking into account its radius and overall mass distribution. You will learn to deal with these issues more precisely later in the course. In this experiment,




will be held constant while




is varied by shifting mass from one side to the other. The acceleration of the system will be calculated for each case. We will use these results to calculate
and
by solving Equation (8) for
. EXPERIMENT NO. 3 1. To simplify the notations, we will call the left-hand side mass hanger

and the right-hand side mass hanger

. Assemble the equipment in the following way. Place a 



mass on each mass hanger. Add a 


mass to the

side and five


masses to the

side. 2. The pulley used in this experiment is called a "smart" pulley because it can, along with the proper computer interface, allow you to obtain a record of position versus time for the two mass hangers. The pulley works in the following way. The yolk, in which the pulley is mounted, contains a light emitting diode (LED) on one side and a photodetector on the other side. When the spokes of the pulley alternately block and unblock the light, a series of pulses is generated. The distance between the spokes is known, and the time interval between them can be found using the computer's internal clock as a timer. Since the two mass hangers are connected to the string that passes over the pulley, a record of their position versus time can be obtained. Log in to the Student account. From the Start menu, open the 1121 folder and the Exp3– Atwood’s Machine file. 3. Each time before acquiring data, transfer one


mass from

onto

, bring

to the top position, make sure the mass hangers are as stationary as possible (not swinging in any direction), release

, and click Record to record data. Click again on the same icon (Stop) to stop recording data before

reaches the pulley and before

strikes the floor. 4. Your data set contains the linear velocity of the masses

and

as a function of time as they move up (

) or down (

). It is stored as Run #1 and listed under the Data Summary Tools Palette window. To view a graph of your data, click on Data Selector at the graph toolbar menu in your workbook and select Run #1. You should now see a graph of linear velocity versus time. 5. You are now going to use the computer to fit the data. PASCO Capstone will fit the data to the “best” straight line, thereby minimizing the error you might make in eyeballing the best straight line. Such a process is called the least-squares fit. Click Fit on the graph toolbar and select Linear: mx + b. A small window will appear with the values of the fitting parameters. Write the value of the slope in the table. If you want to fit only part of the data, highlight the interval that you want to consider (Please ask the TA to show you how to select the data of interest.) and then use Linear Fit. To obtain a total of five values for




, transfer one


mass at a time from

to

and repeat the measurement five times for each case. Label the graph with your names and values for








and print the graph for each group member containing five runs for each


mass transfer. To print the graph, click on the Print icon on the workbook toolbar. (You may also ask your TA to help you with printing the graph). 6. Record your data in Table 3.1.
































































Table 3.1: Atwood Machine - Data








































7. Use the graph paper provided and plot a graph of

 versus




. Use this graph to determine the acceleration due to gravity g and the frictional force f of the pulley. Show detailed calculations on the graph.

















































Close the program without saving data and log out of the Student Account. QUESTIONS 1. Suppose a student modifies the setup used to study Newton’s laws of motion. The lightweight pulley is replaced with one of mass

and radius
. There is no friction in the system, and the track is horizontal. For these modifications, the acceleration of the cart is given by













, where
and
are the mass of the cart and hanging mass, respectively. A student used several quantities as the hanging mass and measured the acceleration for each case. From her data, she plotted the graph shown below. (a) What is the experimental value of g obtained by the student? (b) The mass of the cart is twice the mass of the pulley. What is the mass of the pulley?