Lab 5

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Atwood’s Machines (Lab 5)
Fariha Newaz
Lab Performed on: February 25, 2019
Report Due: March 4, 2019
Lab Partner: Paulina Staszel
L06
TA: Ning Su
Statement of Objective:
The objective of the lab was to test Newton’s second law and to experimentally measure the acceleration due to gravity with an Atwood’s machine. 
Theory:
The Atwood’s machine was device created by George Atwood in 1784 to calculate force and tension and to verify the laws of motion that are under constant acceleration. 
Description of Experimental Setup/List of Equipment Used
The photogate pulley is connected to the 850 Universal interface which is connected to the computer running the Capstone program. One the photogate pulley is a string and on either side of the string are hooks for the masses. 
Materials Used:
· Computer w/ Capstone Program
· 850 Universal Interface 
· Photogate Pulley
· Masses
Procedure
Once the photogate pulley is connected to the computer and Capstone program, the hardware setup tab is used to calibrate the data and once that has been done, in the display menu, display two graphs. One graph displays a quadratic curve of the motion of the masses on the pulley, and the second graph is a linear graph of the motion. The 2 times the “a” value of the quadratic graph is equal to the slope of the linear graph which is the acceleration of the graph. The acceleration can be found by selecting the upward sloping part of the quadratic graph and display the equation and select the corresponding part on the linear graph and display the linear. Check to see if 2a=m or around the value to see if the run was successful. In the first experiment, the mass 1 and mass 2 values are changed 5 times, but the total mass (M1+M2) is kept constant throughout the 5 sets and the acceleration is recorded each time following the steps described earlier. In the second experiment, mass 1 is held constant and mass to is being increased by an interval of 10g; mass 2 is increased 5 times. The acceleration is recorded every time the mass is increased. 
Data
Table 1
Table 1 includes all the data that was recorded during the first experiment of the lab. 
	Experiment 1 Data
	Mass 1 (kg)
	Mass 2 (kg)
	Acceleration (m/s2)
	0.090
	0.010
	7.55
	
	0.080
	0.020
	5.63
	
	0.070
	0.030
	3.89
	
	0.060
	0.040
	1.9
	
	0.055
	0.045
	0.88
	
Table 1: necessary data for experiment 1 calculations
Table 2
Table 2 includes all the data that was recorded during the second experiment of the lab.
	Experiment 2 Data
	Mass 1 (kg)
	Mass 2 (kg)
	Acceleration (m/s2)
	0.030
	0.050
	2.0
	
	
	0.060
	2.3
	
	
	0.070
	3.23
	
	
	0.080
	3.74
	
	
	0.090
	4.21
	
Table 2: necessary data for experiment 2 calculations
Data Analysis
To find the theoretical acceleration of the masses on the Atwood’s machine in experiment 1, Equation 1 will be used, which can also be written as: 
Equation 1
	Where:
		a= acceleration in (m/s2)
		M1= mass 1 (kg)
		M2= mass 2 (kg)
		g= 9.8 m/s2
Sample Calculation:
Sample calculation for the solve of the theoretical acceleration using the first set of masses from Table 1. 
To find the net force applied on the Atwood’s machine in experiment 1, Equation 2 will be used, which can also be written as:
	Equation 2
	Where: 
		Fnet= the net force applied in Newtons (N)
Sample Calculation:
Sample Calculation for the solve of the net force applied using the first set of masses from Table 1.
Table 3
Table 3 contains the calculated theoretical accelerations and net force using Equation 1 & 2.
	
	Theoretical Acceleration & Force
	
	Mass 1 (kg)
	Mass 2 (kg)
	Theoretical Acceleration (m/s2)
	Force (N)
	
	0.090
	0.010
	7.84
	0.78
	
	0.080
	0.020
	5.88
	0.59
	
	0.070
	0.030
	3.92
	0.39
	
	0.060
	0.040
	1.96
	0.19
	
	0.055
	0.045
	0.98
	0.098
Table 3: All calculated values are listed with correct units
											Figure 1 
Figure 1 is the graph of Force vs Acceleration, the graph is to confirm the relationship between, acceleration, force, and mass in Atwood’s machines.
 
Figure 1: The graph is perfectly linear hence the R2 value being really close to 1; implying minimal error. Experimental acceleration was used to display any errors that could have resulted from the lab.
Figure 2
Figure 2 is the graph the accelerations from experiment 2 vs the (M1-M2)/(M1+M2) using the data from Table 2. 
Figure 2: The line of best fit has a fit value very close to 1 implying minimal error in lab.
Newton’s second law is the base equation used to derive the tension equation that will be used in the calculation for tension. The law can be written as:
Equation 3
					
where:
		= Net Force in Newtons
		M= mass of blocks in kilograms
		a= acceleration in m/s2
Figure 3
Figure 3 is the diagram that will be used to derive the Tension equation for mass 1 and mass 2. 
Figure 3: the tension of the spring will be equal in mass 1 and mass 2 due the pulley being an ideal pulley.
The derivation of the tension of the string can be written as followed and the final equation (Equation 4) will be highlighted.
Equation 4
Mass 1: Mass 2: 
 				
 
 
	Where: 
		T= tension (N)
Since this is an ideal pulley, the tension on the strings are equal in the two masses and so are the magnitudes of the accelerations using the equation for mass 2 is best for experiment two because that is the mass that is changing, thus affecting the Tension.
Sample Calculations
Sample calculation of the Tension on the string in the second experiment using the first Mass 2 from table 2.
					T= .060N
Table 4
Table 4 lists the mass 2 values from Table 2 and the calculated Tension values using Equation 4.
	Tension
	Mass 2 (kg)
	Tension (N)
	0.050
	0.6
	0.060
	0.75
	0.070
	0.91
	0.080
	1.08
	0.090
	1.3
Table 4: Tension values seem to increase as more mass is added
Discussion of Data
When comparing the accelerations determined from the experiment to the accelerations determined through equation 1, there was only a minor difference between the values. The differences most probably resulted from not selecting more precise/correct regions on the graphs of the PASCO program. Figure 1 uses the acceleration data collected during the lab instead of the theoretical accelerations so that the extent of the errors can be seen. The total mass was .1kg therefore the perfect non error slope would also be .1, however since the experimental accelerations were used, the slope was .1035 which is extremely close to .1 therefore the errors in this lab were minimal. However, these deviances also show that the ideal pulley scenario does not hold very well, because factors like the actual weight of the pulley and any friction that occurred could have also lead to the difference between the theoretical and experimental accelerations. The relationship between the acceleration and the masses when the force is held constant is that, the acceleration decreases when constantly decreasing the mass on one and constantly increasing the mass on the other, never interchanging. 
Figure 2 is the graph of the acceleration vs (M1-M2)/(M1+M2) where the slope of the graph should equal to the value of “g” which is the acceleration due to gravity constant equal to 9.8 m/s2. It is can be inferred that the slope should be the value of g by rearranging equation to solve for g which can also be written as: 
This equation can also be written as: 
This equation is also known as “a vs (M1-M2)/(M1+M2)” therefore the slope of the graph is indeed equal to the value of g. The slope of the line in Figure 2 is -9.29, the value of g is 9.8. The difference in the magnitude of the two values could be a result from errors in the second experiment, which are similar to those of the first. The negative value in the slope is due to the direction of the blocks going in the negative direction, causing the acceleration due to gravity to be negative. 
The tensionin the string increases when more mass is added. The tension increases because the mass 2 values are also the added force, the more force added to a system the more tension in the string. 
Conclusion 
The lab was done fairly smoothly and without that many errors. If there were any errors that come out of this lab they were probably from either error in the computer software used, errors in the calculations, or a combination of both. 
References
Lab Manual 5: Experiment 5: Atwood’s Machines.
Force vs Acceleration	
7.55	5.63	3.89	1.9	0.88	0.78400000000000003	0.58799999999999997	0.39200000000000013	0.19599999999999998	9.8000000000000032E-2	Acceleration (m/s2)
Force (N)
Acceleration vs (M1-M2)/(M1+M2)	
-0.25000000000000006	-0.33333333333333331	-0.40000000000000008	-0.45454545454545459	-0.5	2	2.2999999999999998	3.2280000000000002	3.7380000000000004	4.2080000000000002	 (M1-M2)/(M1+M2)
Acceleration (m/s2)