# Linearizing these various relationships gives us a

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Linearizing these various relationships gives us a final equation of: 2
1 t 2 = g 2 h ( M + I r 2 ) ∆m τ 2 hr ( M + i r 2 ) (1) This gives us a linear equation of 1 t 2 vs ∆m with a slope (in yellow) and a y-int (in blue). The slope allows us to calculate g while the y-int allows us to calculate τ . Looking at equation (1), we can see that to determine our two unknowns (g and τ ), we need to find M, r and h. M is simply the total mass of the system: M=m 1 +m 2 + m w , which we can find by measuring each of the masses suspended by the string, and the mass of a single washer. All three of these are found using a scale, and the mass of the washers are taken all together and divided by the # of washers present (10). The height (h) is found by measuring the distance (in m) between the resting point for m 2 , which is the base containing the solenoid, and the bottom of the mass when it is positioned at the top of the system (Figure 1). The radius (r) is found using a Vernier caliper (Figure 2), and is found by measuring the distance between the two strings (the diameter, d) and divided by 2 ( r = d 2 ¿ , ensuring the reading error is considered to ½ the smallest division. It’s important to measure the distance between the strings at the very top near the pulley, not near at the bottom by the weights. This is because the weight of the masses causes the strings to pull inwards, making the distance less than the diameter of the pulley. For m 1 , m 2 , m w , h and r, the average measurement and the statistical error on the average are calculated using three different equations (m 1 is used as an example, though these are used for all 5 values); m 1 av = m 1 a + m 2 b + m 3 c N (2) σ IS = m max m min √ N (3) σ mean = σ IS N (4) 3
Where X max and X min are the highest and lowest measurements of the dataset, and N is the number of trials used. The inefficient statistics (IS) equation is used when the data set is small (less than 10 trials). To determine which error to use with the final average measurement, the following test is used: If σ IS ( 2 X ) ,use the Reading Error ( ) If σ IS ( 2 X ) ,use σ mean To create a plot of 1 t 2 vs m, we need to determine the time (t) it takes for m 1 (the lighter mass) to rise a distance (h). Atwood’s machine, controlled by a trigger switch that automatically releases m 1 and simultaneously starts the clock measures t. Five trials will be taken to ensure an accurate representation of t, excluding any possible outliers. Our average time (T avg ) is calculated by: t avg = ∑t 1 + t 2 + …t x N (5) with an error given by inefficient statistics (Equations 3 and 4). From here, we take the recipricol of t avg squared (our y axis), with an error of: σ ( 1 tavg 2 ) = ( 2 t avg 3 ) σ t avg (7) Our x axis is the difference of mass, calculated for each of the five different configurations: m ( ¿¿ 1 + y m w ) ∆m x y = ( m 2 + x m w ) ¿ (8) with an error of: σ Δm = σ m 1 2 + σ m 2 2 + 100 σ mw 2 (9) Where x is the number of washers on m 2 and y is the number of washers on m 1, The slope of our 1 t 2 vs∆m
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