Atwood's Machine

Atwood’s machine was invented George Atwood, an English mathematician, in 1784. The machine consists of two masses, m₁ and m₂, and a pulley. If m₁ is smaller than m₂ in mass, m₂ will accelerate downward and m₁ will accelerate upward. If the string is tight and does not experience stretching, both m₁ and m₂ will encounter the same and same acceleration and can be considered a single system. The acceleration of the machine is considered based on m₁ and m₂, from both the difference and the sum of the two masses. The main purpose of this project is to discover how mass plays a factor in acceleration in Atwood's Machine. The driving question is how is the acceleration of the two masses of an Atwood's machine affected by their difference in mass and by their total mass? Our hypothesis is if mass one is two times smaller than mass 2 then the acceleration will vary greatly.

• Data collection system
• Smart Gate photogate
• Super Pulley with Mounting Rod
• Mass and Hanger Set
• Table clamp or large base
• Support rod, 60 cm or taller
• Right angle clamp
• Thread, about 1m
• Scissors

Part 1 - Varying Mass Difference

Set up

1. Assemble and mount the table clamp and support rod on the edge of a table, or assemble the base and support rod and place it on the tabletop.

2. Attack the pulley to the tab of the photogate using the mounting rod so that the spinning pulley spokes will interrupt the beam of the photogate.

3. Attach the mounting rod horizontally on the support rod using the right angle clamp. Place it near the top of the rod with the pulley over the tabletop.

4. Cut a length of thread about 15 cm longer than the distance from the top of the pulley to the tabletop.

5. Place the thread on the pulley, threading one end through the gap between the pulley and its frame. Tie a loop on this end of the thread and place an empty mass hanger on it. Rest the attached hanger on the tabletop.

6. Tie a loop on the other end of the thread just below the pulley. Place the other empty mass hanger on this loop.

7. Connect the photogate to your data collection system.

8. Configure the data collection system for a Photogate with Pulley or Smart Pulley (Linear) with the default spoke arc length of 0.015 m to measure the linear speed.

9. Create a graph display of Linear Speed on the y-axis with Time on the x-axis.

10. Add 140 g of mass (suggested masses: 100-g + two 20 g) to the 5-g hanger resting on the tabletop, for a total of 145 g hanging from the string on the side of the pulley closer to the mounting rod.

NOTE: Throughout this activity, the mass hanging on the side of the pulley closer to the mounting rod will be the lesser of the Atwood's machine's two masses and will be referred to as Mass 1.

11. Support the suspended hanger with your hand to prevent it from dropping as you add the following masses to it. Add 195 g of mass (suggested masses: 100-g + 50-g + 20-g + 10-g + three 5-g) to the 5-g suspended hanger, for a total of 200 g hanging from the thread on this side. Continue to support it with your hand.

NOTE: Throughout this activity, the mass hanging on the side of the pulley farther from the mounting rod will be the greater mass and will be referred to as Mass 2.

12. Before you collect data, practice releasing and catching the masses: Slightly lower Mass 2 so Mass 1 lifts just off the tabletop. Once any swinging has settled, release Mass 2 and then gently catch the rising mass just before it strikes the pulley. Once you are practicing, return Mass 1 to the tabletop and continue holding the greater mass suspended just below the height.

Collect Data

13. Begin data recording.

14. Release Mass 2 and then catch the rising Mass 1 just before it strikes the pulley.

15. Stop data recording.

NOTE: If the two mass hangers collided, delete the run and record another.

16. Gently lower Mass 2 to rest on the tabletop.

17. Use the tools on your data collection system to determine the slope of a linear fit to your Linear Speed versus Time data during the time when the masses were moving freely. Record this as the acceleration of the system in Table 1.

18. Also in Table 1, record the mass of Mass 1 (including the hanger) and that Mass 2 (including the hanger).

19. Repeat data collection four more times, transferring 5 g from Mass 2 (the greater mass) to Mass 1 between each trial.

Part 2 - Varying Total Mass

Collect Data

20. Continue with the Part 1 setup for Part 2.

21. Copy the data from your final trial (Trial 5) in Part 1 into the first row of Table 2.

NOTE: This eliminates the need to report data collection for this same mass combination in Part 2.

22. Remove 30 g from each of the mass hangers, leaving a total of 135 g (including hanger mass) on Mass 1 and 150 g (including hanger mass) on Mass 2.

23. Repeat the data collection steps from Part 1 to determine the system's acceleration. Record the acceleration in Table 2 as Trial 6. Also record the masses Mass 1 and Mass 2.

24. Repeat data collection four additional times, removing an additional 30 g from both sides between each trial.

• Handle all the equipment with care
• Do not put any additional body parts, expect your hands, near the experiment while the object is at motion, for the object in motion may be heavy and fast.
• Make sure that the experiment in on a secure surface
• Adult supervision is required.

GRAPHS

OBSERVATIONS

The acceleration decreases in varying total mass unlike the acceleration of varying mass difference. The acceleration of varying mass difference increased as the mass difference increased.

ANALYSIS AND DISCUSSION

Masses affects acceleration as a whole. In an Atwood's machine, mass and acceleration go hand in hand and go into the equation. mtotal*a = m2g-m1g has masses on both sides which influence how the results will output.

CONCLUSION

Overall, Atwood's machine is highly dependent on acceleration. Acceleration can increase and decrease but overall throughout the project acceleration decreased as both masses increase and decrease.

The Atwood machine can be an example of elevators, garage doors or wells. For an Atwood's machine is simply two masses hanging over a pulley. So when the masses are not equal then the system of masses will accelerates, causing the masses to be in motion, which is how an elevator is constructed. The Atwood machine helps our society by change the direction of a force, which can make it much easier for us to move something, like opening a garage door or getting water from a well. Such information can help in future by showing people, what two mass can give a correct acceleration, like how fast a door opens and etc. Thus, trying this experiment with different wheels and different mass, can help answers more questions in the future.

1. For each part of your experiment, list each variable and indicate whether it was held constant, increased, or decreased.

• The acceleration increased and the total mass remained constant. Mass 1 increased while mass 2 decreased, lowering the mass difference each trail.

2. How did changing the difference in mass between the two sides affect the acceleration of the Atwood's machine?

• As mass 2 gave 5 kg to mass 1, the acceleration of Atwood's machine increased throughout each trial.

3. Based on your data, express the relationship between the acceleration, ay and mass difference m2-m1 by completing this proportionality statement:

• ay ∞ ____M2__________ (total mass held constant)

4. How did changing the sum of the two hanging masses affect the acceleration of the Atwood's machine?

• The acceleration lowered as the mass went down.

5. Based on your data, express the relationship between the acceleration ay and total mass m2 + m1 by completing this proportionality statement:

• ay ∞ ______M1_________ (mass difference held constant)

6. Combine the two relationships above into a single proportionality expressing the relationship between the Atwood's machine acceleration ay the mass difference m2-m1 and the total mass m2+m1:

• ay ∞ _____M2 + M1____________

7. Convert the proportionality statement above into an equation by introducing a proportionality constant k:

• ay = __P*V________________

8. Use your data to determine the proportionality constant k. Be sure to specify its units. Briefly explain how you determined its value and units.

• The k constant can be determined by taking the average of each P x V value from each ordered pair so

Data Table 1- proportionality constant k is 1.35 m/s^2

k=(.78 +.80+1.08+1.47+2.62)/5 = 1.35 m/s^2

Data Table 2- proportionality constant k is 1.47 m/s^2

k=(2.62 +2.27+1.61+1.24+.75+.35)/6 = 1.47 m/s^2

9. Consider this free-body diagram of an Atwood's machine. Assume that the masses of the string and pulley are negligible. The analysis can be simplified if the system is defined to consist of the two masses linked together, as indicated by the dashed line. This allows you to disregard the string tension as an internal force and consider only the two forces m1g and m2g acting on the system. You can also consider the system to be moving in one dimension, with positive defined in the direction of m1 ascending and m2 descending, as indicated. Apply Newton's Second Law, a = F net / m, to derive an expression for the acceleration ay of the system in terms of the masses m1 and m2.

• mtotal*a = m2g-m1g

10. How does the expression for acceleration that you determined from your data analysis compare to the equation derived above from the Newton's Second Law? Justify your answer.

• The expression is a very simple to Newton's Laws of Motion, For Newton's Second Law of Motion says that the force required to move something equals the object's mass times it's rate of acceleration: F = ma.

SYNTHESIS QUESTIONS

1. One way to check whether a derived relationship is reasonable is to consider whether it behaves as expected in extreme or limiting cases. Determine whether the relationship you derived between the acceleration a of an Atwood's machine and its two hanging masses m2 and m1 reduces to a reasonable form when the two masses are equal. Explain your reasoning.

• The acceleration increases when the two hanging masses m2 and m1 are reduces to a reasonable form when the two masses are equal.

2. Similarly, determine whether the relationship you derived between the acceleration a of an Atwood's machine and its two hanging masses m2 and m1 reduces to a reasonable form when the mass m2 is much greater than m1. Explain your reasoning.

• The acceleration decreases when two hanging masses m2 and m1 reduces to a reasonable form when the mass m2 is much greater than m1.

3. A planetary rover carries an Atwood's machine with 100 g one side and 110 g on the other. If the system's acceleration is measured to be 0.176 m/s^2 on a certain planet, what is the acceleration due to gravity on that planet? Which planet in the solar system is it?

• a) Fnet= m2g-m1g

mtotal*a = m2g-m1g

(.100+.110)(.176)= 110g-100g

g= 3.75

b) A planet with that gravity is Mercury

4. What ratio of masses m2/m1 would produce an Atwood's machine whose acceleration is half that of an object in free fall?

• If is acceleration is

mtotal*a = m2g-m1g

7a=5(9.81)-2(9.81)

a=4.204

So half of 4.204 is 2.102. Then

7a=1/2(5(9.81)-2(9.81))

Thus, a= 2.102, so the ratio of masses of mass would be (1/2* m2g/m1g)

5. Elevators cars have counterweights to reduce the amount of work motors need to do to lift the car. You might idealize an elevator system without its motor as an Atwood's machine. If a particular elevator's counterweight mass is 1,000 kg and its elevator car and passengers have a combined mass of 1,200 kg, what acceleration would the passengers experience if the motors and safety braking mechanisms failed? If the elevator car accelerated from rest at a height of 12 m above the ground floor, how long would it take for the car to reach the ground floor? What would be its speed on impact?

• a) Fnet= m2g-m1g

mtotal*a = m2g-m1g

2200a = (1200*9.81) - (1000*9.81)

2200a = 1962

a= 1.12m/s^2

b) x = Vo(t) + 1/2*a*t^2

12= 0+1/2 (10.92)*t^2

t= 1.48s

c) d= Vit + 1/2*a*t^2

12= x+1/2*(1.12)(1.48)^2

12= x+1.23

x= 10.77 m/s