Atwood’s Machine

Introduction

Accurately measuring the acceleration of a falling object due to gravity is difficult because a falling object travels short, laboratory-scale distances very quickly. However, by slowing down an object’s fall according to the rules of Newton’s second law of motion, the acceleration due to gravity can be accurately determined by an indirect method. A device that is used to indirectly measure the acceleration due to gravity is called an Atwood’s machine.

Concepts

• Atwood’s machine
• Rotational motion
• Accuracy and precision
• Acceleration due to gravity
• Friction
• Newton’s second law of motion

Materials

Safety Precautions

Please follow normal laboratory safety guidelines. Make sure the weights are tightly secured to the pulley cord so they will not come loose during the experiment. If one weight falls off one end of the pulley system, one side will crash down to the ground while the other side flies up. The quickly falling and/or rising masses could cause injury. The masses may also be damaged if they hit the floor too hard.

Prelab Preparation

1. Obtain the Atwood’s machine, support stand, support stand clamp, pulley cord, weight set (or equivalent), soft towel and heavy books.
2. Set up the apparatus on at least a 1.5-m high platform as shown in Figure 1. Place heavy books on the base of the support stand to prevent it from tipping when the masses hang over the edge of the platform.
3. Securely tie one 200-g mass to each end of the pulley cord so the masses are balanced and do not fall. The pulley cord should be long enough to allow one mass to rest on the ground while the other mass hangs just below the Atwood’s machine pulley. Make sure the masses are free to move up and down without interference.
4. Place a soft towel beneath the Atwood’s machine to “catch” the masses and prevent them from being damaged when they hit the floor.
5. Draw a free-body diagram of Atwood’s machine and the masses on the blackboard.
6. Discuss the purpose of Atwood’s machine before performing the demonstration.

Procedure

Simple Atwood’s Machine

1. Add 20 grams to one of the 200-g masses.
2. Allow the 220-g mass to fall to the ground. If the 220-g mass does not reach the ground, adjust the height of the clamp holding the Atwood’s machine until the 220-g mass touches the ground.
3. Raise the 220-g mass up to just below the Atwood’s machine pulley (see Figure 1, which is not to scale).
   {13926_Procedure_Figure_1}
4. Ask a student volunteer to use two meter sticks, or a tape measure, to measure the height above the ground from which the 220-g mass will be released (measure from the bottom of the mass). Make a note of the mass position in relation to the pulley so that the release height will be the same for every trial.
5. Obtain a digital timer or stopwatch.
6. Position the 220-g mass at the appropriate release height and make sure the 200-g mass is not swinging (see Figure 1).
7. When the 220-g mass is at the correct height, and the 200-g mass is stationary, simultaneously release the 220-g mass and begin timing with the digital timer.
8. Stop timing when the 220-g mass just touches the floor.
9. Repeat steps 6–8 ten to fifteen times until the timing measurements are consistent.
10. Have the students review the timing data for all the trials and determine the best average time of descent. (Sample methods for determining the best average time (e.g., average all the data, ignore data points like the high and low value then calculate the average, draw a graph and determine a line of best fit).
11. Calculate the acceleration due to gravity (g) using Equation 5 from the Discussion section.
12. Calculate the percent error compared to the literature value of g, 9.81 m/s² (at sea level).
13. Ask the students to list possible sources of error and their effects on measuring the acceleration due to gravity. Was everything accounted for in Atwood’s equation (Equation 5) to accurately determine the acceleration of gravity using the Atwood’s machine? Is there friction in the system? Was rotational motion included?

Accounting for Friction and Pulley Mass

The results from the Simple Atwood’s Machine Procedure are generally not as good as expected (about 30–40% error). This is because the simplified Atwood’s equation (Equation 5 from the Discussion section) assumes the pulleys are massless and frictionless, and that the mass of the string is negligible. However, the results indicate this is obviously not the case. Better results are obtained from the same data by using a more realistic Atwood’s equation (Equation 10 from the Discussion section). Equation 10 accounts for the mass of the pulley sheave as well as the friction in the axles of the pulley.

1. Determine the amount of friction in the pulley system by securely tying a 200-g mass (or equivalent) on both ends of the pulley cord.
2. Add a 10-g mass to one of the 200-g masses.
3. Give the 210-g mass a small tug downward. The tug should be hard enough to just overcome the static friction in the system. How far do the masses travel? Do they stop? Does the heavier mass continue to travel all the way to the floor even with the slightest tug?
4. If the masses stop, add 5 more grams to the 210-g side and repeat step 3. If the masses continue to move to the floor even after the slightest tug, continue to step 5. If the masses stop before reaching the floor repeat step 3 with 5 more grams.
5. Once the masses continue to move all the way to the floor after the smallest tug, remove the last mass (5-g or 10-g) that was added.
6. Add one 1-g mass at a time to the heavier side and repeat step 3. Adding small paper clips to the hanging masses works well if 1-g masses are not available.
7. Repeat step 6 until the smallest amount of extra mass is added that will just keep the masses moving all the way to the floor after a small tug.
8. Measure and record the total extra mass added in steps 3–7. (Use a balance if necessary.) This extra mass is the “frictional” mass, or the mass that is needed to just overcome the torque created by kinetic friction in the pulley axles. Note: The frictional force on the pulley axles increases as more mass is added over the pulleys. However, for this experiment, the difference between the total mass used when determining the amount of friction in the pulley system and the total mass used in the actual experiment is small so any changes in frictional force will be minimal.
9. Use Equation 10 to determine the acceleration due to gravity using the original time data. Note: The mass of the pulley, on average, is 5.3 grams.
10. How close is the “realistic” value to the accepted value of g? What is the percent error compared to the accepted value of g?
11. List other possible sources of error that were not accounted for.

Teacher Tips

• Use the Atwood’s machine when studying constant acceleration, free-body diagrams, Newton’s second law of motion, or when measuring the acceleration due to gravity.
• This economy choice Atwood’s machine, and simple friction-determining procedure, can produce results within 4% of the actual acceleration of gravity. The accuracy of determining the acceleration of gravity can be improved by using better equipment, graphical analysis, and/or photogate timers. However, even with more costly professional-type Atwood’s machines and timing devices, and time-consuming graphical analysis, it is still difficult to measure the acceleration due to gravity to within 2% of the actual value.
• Timing error is a significant source of error in this experiment. The rate of descent is fast, and therefore it is difficult to start and stop a digital timer at the exact instant the mass is released and when it hits the floor. It is best to perform at least 15 trials with the same mass combination so there is plenty of data to review. Perform a few practice releases before the “real” trials to get a feel of when to start and stop the timer. Depending on your confidence of these timing values, they can be used or left out of the final data. After the trials, ignoring the high and low values before calculating the average time of descent may also lead to a better results. Static friction, stretch in the pulley cord, the mass of the pulley cord, and air resistance are other sources of error in the experiment that may be discussed.
• Using the Vernier Smart Pulley (Flinn Catalog No. TC1220) and Vernier Photogate (Flinn Catalog No. TC1517) can produce results closer to 2% of the accepted value of the acceleration due to gravity, g. See Vernier’s lab manuals Physics with Computers (Flinn Catalog No. TC1331) for more information on using a photogate timer and pulley as an Atwood’s machine.
• Students can try a lighter or heavier total mass as well as smaller or larger mass differences between the hanging masses to see if better results are achieved. Typically, the heavier the total mass and the greater the mass difference between the two masses, the better the accuracy of the results. As the total mass increases, the mass of the pulley and the mass of the pulley cord become less significant. The frictional force increases as the total mass increases, but this can be measured and accounted for. When there is a large difference between the hanging masses, the static friction in the pulley axle is more easily overcome at the start of the experiment. However, the larger the difference in mass between hanging masses, the faster the masses will accelerate, making it more difficult to time accurately. Increasing the height of the platform may help to improve timing accuracy if the acceleration is too fast.
• Tying extra pulley cord to the bottom of both masses so that the two ends of the pulley cord always touch the floor no matter how high the masses are will eliminate any possible errors due to the pulley cord mass moving from one side of the pulley system to the other.
• Tall filing cabinets make great tall platforms for this demonstration.

Sample Data

Release height (h): 163.5 cm
Mass 1 (m₁): 200 g
Mass 2 (m₂): 220 g
Average time (t): 3.55 s
Acceleration (a) = (2 x 163.5 cm)/(3.55)² = 25.9 cm/s² = 0.259 m/s²

Determining the acceleration due to gravity

Simple Equation (Equation 5)
g = 0.259 m/s² x (200 g + 220 g)/(220 g – 200 g) = 5.44 m/s²

Percent error from the accepted value of g:
[(9.81 m/s² – 5.44 m/s²)/9.81 m/s²] x 100% = 44.5% error

Accounting for Friction and Pulley Mass (Equation 10) 
“Frictional” mass (measured) (m_f): 8.3 g

Mass of pulley sheave (m_s): 5.3 g
g = 0.259 m/s² x (200 g + 220 g + 5.3 g)/(220 g – 200 g – 8.3 g) = 9.41 m/s²

Percent error from the accepted value of g:
[(9.81 m/s² – 9.41 m/s²)/9.81 m/s²] x 100% = 4.1% error

Discussion

The Atwood’s Machine was developed by Reverend George Atwood in the late 18th century to indirectly measure the acceleration of gravity. When two unequal masses are tied to the ends of a length of string and hung over a pulley, the larger mass will accelerate down, while the smaller mass accelerates up at the same rate. If the pulleys are assumed to be massless and frictionless, the acceleration of both masses is the same and it depends on the acceleration due to gravity, as well as the total mass of the system and the difference in mass between the two hanging masses. This follows Newton’s second law of motion, force equals mass times acceleration (F = ma). Refer to Figure 2 and the following equations for the derivation of the theoretical Atwood’s machine equation.

Where:

T₁ = tension in string 1
T₂ = tension in string 2
m₁ = mass 1
m₂ = mass 2 (m₂ > m₁)
a = acceleration of both masses
g = acceleration due to gravity

Equation 5 is a theoretical formula for determining the acceleration due to gravity from Atwood’s machine. In the classroom however, pulleys will always have mass and be slowed by friction. Because the pulleys in the Atwood’s machine have mass, when they rotate their rotational energy takes away from the total energy that would otherwise be used to move the masses (in the ideal situation). Therefore, the observed acceleration of the masses is lower than expected assuming the pulleys were massless. The pulley sheave axles also produce a frictional force that acts against the rotation and decreases the total energy of the system. The frictional force is not constant, but increases as more weight is placed on the axles. In order to obtain realworld results that are closer to the accepted value of gravitational acceleration, the Atwood equation needs to include both the mass of the pulley as well as a term to account for the friction in the pulleys.

A rotating object experiences a torque that is equal to its rotational moment of inertia times its angular acceleration (τ = Iα). A torque is equal to a force times the distance of the lever arm where the force is applied (τ = F x r). In the case of Atwood’s machine, the lever arm is the radius (R) of the pulley sheave (see Figure 3). Therefore, the torque is equal to the net force on the outer rim of the pulley sheave times the radius, R, of the pulley. Rotational moment of inertia is equivalent to the “rotational” mass of the object about an axis of rotation. Assuming the pulley sheave is a solid cylinder with mass m_s and radius R, its rotational moment of inertia about the center is equal to ½m_sR². Angular acceleration is simply the linear acceleration divided by the radial distance from the center of rotation (α = a/R). The friction along the axle of the pulley sheave produces a torque that acts against the rotation. Since the torque produced by friction acts against the motion, it can be assumed that an additional mass has been added to the smaller mass which causes the acceleration of the masses to be less than expected compared to the ideal situation. The torque produced by this “frictional” mass, m_f, is equal to m_fgR. Since the “frictional” mass is a measure of the total friction produced by the two-pulley Atwood’s machine in this experiment, it is assumed that each pulley produces half the total measured amount of friction.

The net force on the outer edge of the pulley sheave is therefore: Equation 7 reduces to: Substitute for T₁ and T₂ from Equations 1 and 2: Rearrange to solve for g: The pulley sheave’s mass is a constant. For the Atwood’s machine in this experiment the mass of the pulley sheave is 5.3 ±0.1 grams. The “frictional” mass will be measured during the experiment and is dependent on the total mass hanging from the pulleys. The masses acceleration is determined by the following equation:

h = distance the masses move (heavy mass falls, lighter mass rises)
t = time it takes the masses to travel distance h

References

“Making Atwood’s Machine ‘Work.’” The Physics Teacher, Vol. 39. March 2001, pp. 154–158.