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

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)

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

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

Hypothesis   T T M1 M2 M1 g M2 g

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

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

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

Raw and Processed Data for 100 g. mass (constant for all trials) Raw Data Trial Constant Mass (kg) (+/-) kg (+/-) % 1 0.1000 0.00005 0.0500% 2 3 Processed Data Average 0.1000 0.00005 0.0500% Stdev 0.0000 0.00000 0.00

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 0.00005 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 0.00005 0.0100% 6.339 0.001 0.016% 0.9957 Stdev 0.0000 0.00000 0.00 0.143 0.0004

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 0.00005 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 0.00005 0.0250% 3.109 0.001 0.032% 0.9997 Stdev 0.0000 0.00000 0.00 0.014 0.000 1.4E-06 0.0001

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 0.00005 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 0.00005 0.1000% 3.040 0.001 0.033% 0.9997 Stdev 0.0000 0.00000 0.00 0.007 0.000 0.0 0.0001

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 0.00005 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 0.00005 0.1667% 4.920 0.001 0.020% 0.9998 Stdev 0.0000 0.00000 0.00 0.017 0.000 0.0001

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 0.00005 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 0.00005 0.2500% 6.132 0.001 0.016% 0.9998 Stdev 0.0000 0.00000 0.00 0.021 0.000 0.0001

Mass M2 vs. System Acceleration

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

Difference in Masses/Sum of Masses vs. System Acceleration

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.

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

Sample Calculation m/s/s

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%

Theoretical and Experimental Values

500g constant weight collected data

Collected data for 500g versus weight added

300g constant weight collected

Collected data for 500g versus weight added

Percent uncertainties versus weight added

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

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

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