1. By: Dylan Van de Kerkhove Jack Pfister Lucas Mang Zach Simons
Investigating the Effect of Mass on the Acceleration of an Atwood's Machine
By: Dylan Van de Kerkhove
Jack Pfister
Lucas Mang
Zach Simons

2. Abstract
Determined theoretical acceleration depending on the 2 masses of an Atwood’s Machine to be
Measured acceleration with 0.1 kg mass and 5 other masses using a photogate (by finding slope of velocity vs. time graphs)
Experimental values were close to but lower than theoretical, mainly due to friction that was unaccounted for in theoretical calculations
As difference in mass approaches (+/-) infinity, acceleration of the system approaches (+/-) 9.8 m/s/s asymptotically (respectively)

3. Research Question
How does changing the mass of one side of an Atwood’s machine affect the acceleration of the system?

4. Hypothesis -a +a Mass M1 > Mass M2 To solve equation Action Force
Reaction Force
M1g (Earth pulls down on M1)
M1 pulls up on Earth
M2g (Earth pulls down on M2)
M2 pulls up on Earth
String pulls up on M1 and M2
M1 and M2 pull down on string M1 M2 -a +a

5. Hypothesis T T M1 M2
M1 g
M2 g

6. Hypothesis
Graph of predicted accelerations (negative is acceleration in direction of m2)
y(x)= (9.8)×[(0.1-x)÷(0.1+x)]

7. Materials Two Vernier pulleys Post to hold up pulleys at even heights
Table
String with minimal mass
Differing masses (we used two 100 g masses, and one each of 200g, 500g, 50g, 30g, and 20g)
Photogate connected to laptop with motion sensor software

8. Procedure (Videos in file)
We are actually finding the slope of the tangent line of the part where the mass is falling, not taking the derivative
The photogate is set up so that it reads the amount of time it takes for the beam to reconnect in the next gaps between spokes in the pulley

9. Raw and Processed Data for 100 g. mass (constant for all trials)
Raw Data
Trial
Constant Mass (kg)
(+/-) kg
(+/-) % 1
0.1000
0.0500% 2 3
Processed Data
Average
0.1000
0.0500%
Stdev
0.0000
0.00

10. 500 g. Mass Raw and Processed Data
Raw Data
Trial
Independent Mass (kg)
(+/-) kg
(+/-) %
Velocity Equation Slope (m/s/s)
(+/-) m/s/s r 1
0.5000
0.0100%
6.174
0.001
0.016%
0.9954 2
6.417
0.9955 3
6.425
0.9962
Processed Data
Independent Mass (kg)
(+/-) kg
(+/-) %
Velocity Equation Slope (m/s/s)
(+/-) m/s/s r
Average
0.5000
0.0100%
6.339
0.001
0.016%
0.9957
Stdev
0.0000
0.00
0.143
0.0004

11. 200 g. Mass Raw and Processed Data
Raw Data
Trial
Independent Mass (kg)
(+/-) kg
(+/-) %
Velocity Equation Slope (m/s/s)
(+/-) m/s/s r 1
0.2000
0.0250%
3.099
0.001
0.032%
0.9996 2
3.104
0.9998 3
3.125
Processed Data
Independent Mass (kg)
(+/-) kg
(+/-) %
Velocity Equation Slope (m/s/s)
(+/-) m/s/s r
Average
0.2000
0.0250%
3.109
0.001
0.032%
0.9997
Stdev
0.0000
0.00
0.014
0.000
1.4E-06
0.0001

12. 50 g. Mass Raw and Processed Data
Raw Data
Trial
Independent Mass (kg)
(+/-) kg
(+/-) %
Velocity Equation Slope (m/s/s)
(+/-) m/s/s r 1
0.0500
0.1000%
3.041
0.001
0.033%
0.9998 2
3.033
0.9997 3
3.046
Processed Data
Independent Mass (kg)
(+/-) kg
(+/-) %
Velocity Equation Slope (m/s/s)
(+/-) m/s/s r
Average
0.0500
0.1000%
3.040
0.001
0.033%
0.9997
Stdev
0.0000
0.00
0.007
0.000
0.0
0.0001

13. 30 g. Mass Raw and Processed Data
Raw Data
Trial
Independent Mass (kg)
(+/-) kg
(+/-) %
Velocity Equation Slope (m/s/s)
(+/-) m/s/s r 1
0.0300
0.1667%
4.910
0.001
0.020%
0.9997 2
4.940
0.9998 3
4.911
Processed Data
Independent Mass (kg)
(+/-) kg
(+/-) %
Velocity Equation Slope (m/s/s)
(+/-) m/s/s r
Average
0.0300
0.1667%
4.920
0.001
0.020%
0.9998
Stdev
0.0000
0.00
0.017
0.000
0.0001

14. 20 g. Mass Raw and Processed Data
Raw Data
Trial
Independent Mass (kg)
(+/-) kg
(+/-) %
Velocity Equation Slope (m/s/s)
(+/-) m/s/s r 1
0.0200
0.2500%
6.145
0.001
0.016%
0.9997 2
6.108
0.9999 3
6.144
Processed Data
Independent Mass (kg)
(+/-) kg
(+/-) %
Velocity Equation Slope (m/s/s)
(+/-) m/s/s r
Average
0.0200
0.2500%
6.132
0.001
0.016%
0.9998
Stdev
0.0000
0.00
0.021
0.000
0.0001

15. Mass M2 vs. System Acceleration

16. Difference in Mass (M1 - M2) vs. System Acceleration

17. Difference in Masses/Sum of Masses vs. System Acceleration

18. Qualitative Observations
For most trials, the string would fall off of the pulley tracks at some point during the fall.
After impact of heavier masses, lighter masses would fly up with high velocity away from the system, sometimes into the pulley apparatus.

19. Analysis Forces Acting on m1 Fx Fy -T +m1g Forces Acting on m2 Fx Fy-T
+m1g
Forces Acting on m2 Fx Fy +T
-m2 g

20. Sample Calculation
m/s/s

21. Theoretical and Experimental Values
m1 (kg)
m2 (kg)
Theoretical accel. (m/s/s)
Experimental accel. Averages (m/s/s)
Percent Error
.1000
.5000
-6.533
-6.339
2.970%
.2000
-3.267
-3.109
4.836%
.0500
3.267
3.040
6.948%
.0300
5.277
4.920
6.765%
.0200
6.533
6.132
6.138%

22. Theoretical and Experimental Values

23. 500g constant weight collected data

24. Collected data for 500g versus weight added

25. 300g constant weight collected

26. Collected data for 500g versus weight added

27. Percent uncertainties versus weight added

28. Conclusion (Result Meaning)
As difference in mass approaches infinity, acceleration of the system approaches 9.8 m/s/s asymptotically
As difference in mass approaches negative infinity, acceleration of the system approaches -9.8 m/s/s asymptotically
If one mass equals 0 kg, acceleration will equal 9.8 m/s/s
If (difference in mass/sum of masses) is graphed against acceleration, resulting graph is linear with slope of g
Sum doesn’t always directly affect acceleration, but difference does

29. Conclusion (Sources of Error)
Friction
Between pulley and axel
Caused by weights of masses pushing down on instruments, & b/c pulleys were not 100% in line with each other
Necessary between string & pulley; if static friction is too small or overcome (b/c weights of masses too small or too large), acceleration reading is too small
Air resistance
Larger masses have larger surface areas, increasing drag forces & decreasing resulting system acceleration
Moment of inertia of pulleys (starts motionless & resists rotational motion)
Pulleys are not massless, therefore contribute to upward forces on masses & resist gravitational force
String slipping off pulleys reduces acceleration reading

30. Conclusion (Further Experimentation)
Increased # of pulleys would decrease normal force on each pulley (decreasing upward force on masses & friction of pulleys on axels)
Greasing axels would also reduce unwanted friction
Pulleys with deeper tracks would reduce frequency of problematic slippage of string
Performing experiment in a vacuum would eliminate air resistance