Lab Report 2_ Determining g on an incline

Investigating the Acceleration due to Gravity on an Inclined Plane PCS 120

Introduction The objective of this experiment is to determine the value of g by investigating the acceleration due gravity at several different inclines. The core data collection of the investigation will be recorded using a motion sensor and the software Logger Pro which will graph the movement of the vernier cart on the incline at different angles. Theory When an object is moving due to the influence of the force of gravity, it is considered that the object is in free fall. According to Galileo, any two objects in free fall, regardless of their masses will have the same acceleration (Knight, 2016). The magnitude of free fall is denoted by a 𝑎 → special symbol g , a value which by definition is always positive. The value of g simply represents magnitude, it is oftentimes referred to as the acceleration due to gravity; depending on the situation, a y = a free fall = -g , on earth g = 9.8m/s 2 . When further elaborating on motion on an incline plane as represented in Figure 1 , the free fall acceleration is broken into two pieces as 𝑎 → free fall will just point straight down. The two vectors include ll and the vector ⟂ , in this case 𝑎 → 𝑎 → vector parallel to the incline will be responsible for accelerating the object. The magnitude of 𝑎 → free fall is g , and ll is the opposite of (angle formed by the incline) therefore the magnitude of 𝑎 → θ 𝑎 → ll must be g sin (Knight, 2016). Therefore, the acceleration along an incline is represented by θ the equation a s = g sin . In addition, the force of gravity acting on the vernier cart on an incline, θ air resistance or friction in this case will have a very small impact on heavy objects resulting in slight error explaining why the magnitude of g may not be exactly 9.8 m/s 2 but relatively close to the calculated 9.8 m/s 2 . 1

Figure 1: Average acceleration of each trial plotted with its corresponding angle Trial h 1 (cm) h 2 (cm) Δh(cm) x(cm) Angle(°) Sample Calculations 1 5 12 7 102 3.935 Δh= h 2 /h 1 = 12/5 2 5 16 11 102 6.190 = 7 3 5 19 14 102 7.889 sin θ = opp/hyp = sin^-1(7/102) 4 4.7 22.7 18 102 10.164 = 3.935 5 4.5 28.5 24 102 13.608 Table 2: Calculations for Δ h and θ Table 1: Height, distance, and angle measurements for each trial Trial Angle(°) Average acceleration ( m/s 2 ) Uncertainty (mm) ± Standard Deviation 1 3.935 -0.4078211 0.5mm ± 2.6095952 2 6.190 -0.3629821 0.5mm ± 2.62371619 3 7.889 -0.3024086 0.5mm ± 3.48433073 4 10.164 -0.2318106 0.5mm ± 4.20047869 5 13.608 -0.1382338 0.5mm ± 4.87454672 Table 3: Average acceleration, angle, and standard deviation for each trial 3

The tables and figures above show the general measurements of every trial, as well as one graph showing the relationship between each trial’s average acceleration and their angle. There is a clear increase in the acceleration as the angle increases, which is expected. The end goal is to achieve a value of 9.8 when x= 90. The average acceleration’s were determined by omitting any values before the cart was released and after it bounced off the end stopper. This ensures that the data being analyzed is only from the times of interest as the cart is sliding down the ramp. From there, all the acceleration values were averaged out. Then, the averages were tabulated alongside the angle corresponding to it and it’s standard deviation. The standard deviation for the entire experiment resulted to be 3.5525 by using all the acceleration values from each trial. One method of finding g is by following the formula g= a/sin . An example of this using θ trial 1 would result in a g value of 5.943 m/s 2 . With the standard deviation from trial one, the calculated gravity could lie between 3.3334048 m/s 2 and 8.552 m/s 2 . This is the simplest way to calculate the force of gravity. Below is a more detailed graph of Figure 1 where error trendlines have been added. Theoretically, when the lines reach x=90 representing sin90, the distance between one of the error trendlines and the acceleration trendline should equal 9.8 with standard deviation accounted for. Grey line at sin90 (x=90) is 2.8292 Acceleration trendline at sin90 is 2.0545 Orange line at sin90 is 1.1482 d1= 2.8292 - 2.0545 =0.7747 m/s 2 d2= 2.0545 - 1.1482 = 0.9063 m/s 2 4

References Knight, R. (2016, January 4). Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th ed.). Pearson. 6

Appendices Time Trial 1 Trial 2 Trial 3 Average of trial 1: -0.2403863 m/s 2 0.35 2.64009429 Average of trial 2: -0.167102 m/s 2 0.2 2.60240664 Average of trial 3: -0.0814579 m/s 2 0.45 1.91270139 2 0.5 1.46150818 Average of the averages: -0.1629821 m/s 2 0.55 2.68222824 1.28021574 0.6 1.96558056 2.77231867 0.8447963 Standard deviation of the values: 3.06611217 0.65 1.46076713 2.58361574 0.1474159 0.7 1.25009738 1.81128349 -0.0268366 0.75 0.83669769 1.38311574 0.55361682 0.8 0.15747299 1.21590324 1.10405772 0.85 -0.0636773 0.6803892 1.18435571 0.9 0.51375895 0.01185679 1.0261946 0.95 1.09770586 0.08633225 0.8847071 1 1.19097222 0.68165957 0.831775 1.05 1.05218426 1.09791759 0.83346883 1.1 0.92911713 1.15021451 0.82902253 1.15 0.85956435 1.02545355 0.80197423 1.2 0.82134738 0.91943056 0.72262901 1.25 0.80684398 0.8604642 0.41117654 1.3 0.79366389 0.83357469 -0.7855123 1.35 0.73003951 0.8088554 -3.6587725 1.4 0.43388441 0.78212469 -7.8311452 1.45 -0.7534884 0.70293827 -9.6433815 1.5 -3.6548556 0.36623719 1.55 -8.0988758 -0.9103262 1.6 -10.299528 -3.9618088 1.65 -8.384021 7