Measurement of Earths Gravitational Acceleration … rearranged into the linear equation format ... 4 Discussion −2 … = (−) # $ %% & ' ((> -1, . ) ... Measurement of Earths Gravitational

Benjamen Reed (110108461) [email protected] 1

Determining gravitational acceleration using an inclined air track: Data Handling & Statistics

Benjamen Reed (110108461)

[email protected]

Aberystwyth University

Abstract

The aim of this experiment was to determine the value of gravitational acceleration on the Earth’s surface (gE) using an inclined air track and floating mass. Data analysis techniques were used such as linear regression, error combination and a chi-squared test in order to support the accepted value of gE. The accepted value of gravitational acceleration on Earth’s surface is 9.81ms-2 (3.d.p) and the measured value via experimentation was 10.73±0.32ms-2, which via chi-squared testing, was rejected as supporting evidence. Possible sources of inaccuracy between these values were also discussed.


1 Introduction and Background

Gravity can be defined as infinitely permeating vector field that is produced by mass distorting space-time. A unit mass will experience a force when it is placed in a gravitational field, whilst of course producing its own gravitational field also. This behaviour is analogous to charged particles in an electric field. Gravity is the weakest of the four fundamental forces, but it responsible for the long-distance interaction of planets, stars and galaxies [1]. On Earth, the value of gravitational acceleration at sea level (gE) remains relatively constant at a value of 9.81ms-2 (2.d.p) [2]. An object released from any small arbitrary height on Earth will accelerate toward Earth at this value (assuming no air resistance). Even a glider suspended on an inclined air track will accelerate toward Earth at a rate proportional to this value, although its acceleration in the direction of travel will be somewhat less. Figure 1 shows the forces in action on such a glider, suspended on an inclined air track.

Figure 1 - A diagram showing the forces in action on a car that is suspended on an inclined air track

Using the equations from figure 1, a relationship between the height of the starting position h and the time taken for the glider to cover the distance between two points t, was formulated. The distance between the starting and finishing positions was set as L (i.e. gliders displacement). The application of Newton’s Second Law allowed the conversion of forces into values of acceleration. Newton’s second law states…

“When viewed from an inertial reference frame, the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass.” [3]

…or when represented mathematically… F =m ⋅ a∑

By substituting the equation Fa from the diagram into Newton’s 2nd law and cancelling the mass terms, the acceleration of car can be shown to be…

ax = gsinθ

…or alternatively, by replacing sinθ with h/L (via use of Pythagoras’s theorem)…


ax =ghL (eqn 1)

Equation 1 could have been used to find out the value of Earth’s gravitational acceleration, but the gliders acceleration parallel to the inclined air track had to be calculated. It was far easier to substitute equation 1 into an already established equation of simple kinematics, with height h and time taken t as the variables. The most useful kinematic equation for this purpose is…

Δx =υxit +12axt

2

…where Δx is displacement, υxi is the initial velocity, ax is the acceleration in the direction of motion and t is the time taken to cover the displacement [4]. It was assumed that the initial velocity was going to be zero, which removed the first term on the right-hand side of the equation. Equation 1 was then substituted into ax to give…

Δx = gh2L"

#$

%

&'t2 (eqn 2)

Equation 2 contains two variables (h and t) that could be measured and analysed. An experiment was thus conducted to find the relationship between these variables.

2 Experimental Procedure

The equipment that was used in this experiment is as follows (set up in figure 2):

• PASCO Variable Output Air Supply (SF-9216) • PASCO 2.0 Metre Air Track (SF-9214) + 2 Gliders -13 cm long; 170 g (SF-6306) • 2 PASCO Accessory Photogates (ME-9204B) • PASCO Smart Timer (ME-8930) • Small Variable Hand-Operated Scissor Lift • One metre rule • Soft material to stop glider (i.e. a buffer) • Computer to record data.

Figure 2 - Experiment set-up using equipment from above


The experiment was set up as shown in figure 2. The photogates were placed 1 metre apart to provide a simple displacement value Δx (and L) for the data analysis. At the starting point, the air track was raised by 0.005m, using the hand-operated scissor lift and the one metre rule. The smart timer was turned on, connected to the photogates and set to record the time taken for the glider to travel between the photogates. This would provide our time t values. The air supply was then turned on and the glider was moved to the first photogate, such that the top-leading mark on the gliders mark sheet was just between the photogates sensors (note: the glider was only moved on the track when the air supply was on. Moving the glider when there was no air supply would have possibly caused scratches on the air track which could have affected the experiment). The photogate would only trigger when it detected movement and so the glider was held as still as possible at this position. The smart timer was activated and the glider released from its starting position. The first photogate had a red LED that flashed to signify that it had detected movement. The glider accelerated toward the second photogate. When the top-leading mark on the gliders mark sheet passed the second photogates sensors, the time reading was taken and displayed on the smart timer and hence recorded on an Excel spread sheet. For a height of 0.005m, another four readings were taken, and then the average of the five readings was calculated and recorded. The height of the starting point was then increased by 0.005m and the experiment was repeated. Readings were taken for 0.005m increments all the way up to 0.080m.

3 Data Analysis

For the data analysis, the computational program, SciLab, was used to calculate the value of gravity using the data that was collected from the experiment. The raw data was coded as two row vectors containing 16 elements each; this raw data can be viewed from Table 1, Appendix A. The time-taken t values were then plotted against the height of the cars starting point h. This graph can be seen in figure 3.

Figure 3 - A plot of raw data (t vs. h)


The error in the time values was set at ±0.1s. Although the photogates provided an accurate measurement to within ±0.0001s, this did not compensate for the human error in the starting position. When the height of the starting position was varied, the photogates and the glider had to be moved to maintain a displacement of 1m. It was also difficult the release the glider such that the first photogate activated at a velocity near to zero. Often, the glider would travel a very small distance and gain some velocity before activating the first light gate. Also, the releasing of glider was not always perfect. Sometimes unwanted oscillations would occur as the glider moved due to a small knock or an imperfect release (baring in mind that the glider was released by hand). The error bars showing this can be seen on figure 3.

To perform a regression analysis, the equation and hence data had to be linearised. Equation 2 was rearranged into the linear equation format (y = mx + c), such that…

Δxt2=

g2L"

#$

%

&'h

(eqn 3)

In this situation, y is equal to Δx/t2, the gradient m is equal to g/2L and x is equal to h. The displacement Δx was set at 1m, and so…

h∝ 1t2

In SciLab, the values of t were plugged into the equation…

y = Δxt2 (eqn 4)

…where Δx is equal to 1, in order to provide a row vector of linearized time values (these can be seen in Table 2, Appendix A). The new y-values were then plotted against the h-values to provide a graph showing a linear relationship (see figure 4).


Figure 4 - A plot of linearised data points (y vs. h)

The new error on the y-values was calculated by combining the error on Δx and t. The function for the y-values was equation 4, and this was plugged into the general equation for combining errors. In this case, the equation looked like so…

σ y =∂y∂t"

#$

%

&'2

σ t2 +

∂y∂(Δx)"

#$

%

&'

2

σ Δx2

σ y =4t6×σ t

2"

#$

%

&'+

1t 4×σ Δx

2"

#$

%

&' (eqn 5)

…where σy is the error on y, σ t is the error on t and σΔx is the error on Δx [5]. The error in Δx was set at ±0.005m for this experiment. The error for each y-value was unique because, as shown in equation 5, they depend of the value of t (errors can be seen in table 2, appendix A). Note that the error values on y were too small to be seen in figure 4. Due to each error on y being different, a normal regression analysis would not suffice. A weighting factor had to be taken into account when calculating the gradient and intercept of the linear relationship in figure 4.

A weighted linear regression is used when all the errors on y are different. The best-fit values of gradient and intercept are given by the equations [5]…


m =xy− xyx2 − x 2

c = y −mx

where…

Si =1σ yi2

y = ΣSiyiΣSi

x = ΣSixiΣSi

x2 = ΣSix2i

ΣSi

xy = ΣSixiyiΣSi

By inserting these equations into SciLab and using the linearized data values, the value for gradient was calculated to be 5.365m-1s-2 and the intercept was 0.0043940s-2. For working out the value of gravitational acceleration, only the gradient is considered, and so the error on the gradient can be found using the equation [5]…

σ m =1

x2 − x 2( )ΣSi

Again, by inserting this equation into SciLab and using the linearized data values, the error of the gradient was calculated to be ±0.16m-1s-2. Equation 3 was then taken into consideration, and rearranged so that gravitational acceleration became the subject, such that…

g = 2mL

L is equal to one so the final value of gravitational acceleration was twice the gradient of the linearized data. Therefore, the measured value of g was 10.73ms-2 (2d.p.). To calculate the error of this value, the following equation was used [5]…

σ g = 4σ m2 + 4m2σ L

2

...which gave an error on g of ±0.32ms-2 (2d.p.). Hence the result of this experiment was 10.73±0.32ms-2. All of the SciLab coding used for this analysis can be found in appendix B.


4 Discussion

At the conclusion of the experiment, the value of gravitational acceleration on Earth’s surface was calculated to be…

10.73± 0.32ms−2

As mentioned in section 1, the accepted value of gE is 9.81ms-2 (2d.p). To ascertain whether the calculated value was supportive of gE, a chi-squared test was performed. The null-hypothesis stated that the calculated value of 10.73±0.32ms-2 was supportive of gE. The alternative hypothesis stated the opposite. A confidence of 90% was chosen for the null hypothesis. To create some theoretical values, the accepted gE was divided by two to produce a gradient, and this was used to create a function in SciLab which would create a row vector containing 16 elements that represented a perfect result. These ‘predicted’ values along with the actual measured y-values were plugged into the chi-squared equation [5]…

χ 2 =Σ yobserved − ytheoretical( )

σ yi

#

$%%

&

'((

2

The resulting chi-squared value was 435 (3s.f.). With 16 data points (N) and 2 significant parameters (M), the degrees of freedom n will be (N-M), which is 14. Using a lookup table (see appendix C), the critical chi-squared value was found to be 21.1 (3s.f). Clearly, because χ2 > χ2

critical, the null hypothesis is rejected. Hence, the calculated value of g is not supportive of the accepted value for gravitational acceleration on Earth at sea level.

So, whilst this experiment proved to be fairly precise, it was inaccurate such that it gave a very large chi-squared value, which caused it to be rejected. Such a large inaccuracy in the data means that something systematically was causing the glider to accelerate more rapidly than would be expected if gravity were the only driving force. As discussed in the experimental procedure (section 2), the glider was assumed to be accelerating from an initial velocity of zero. However in practice, the glider would often have to move a small distance before the first photogate activated. Let’s assume for a moment that the air track is inclined so the starting position is 0.05m from the horizontal. This gives acceleration in the direction of motion of about 0.49ms-2 according to equation 1. So in a 1/10th of a second, the glider would have a velocity of 0.05ms-1 (2.d.p). This may not sound like much, but this non-zero initial velocity means that the glider would appear to accelerate faster when observed from the point of view of the photogates, having already gained some kinetic energy; such an error can only be attributed to human error in the positioning of the glider prior to release. Another possible source of error (that would be systematic) is the workbench that the experiment was carried out on. The workbench was assumed to be completely horizontal and so was not taken into consideration when working out the incline of the track. If the workbench had a natural bump on the surface, or was constructed with a slight incline, this could cause the glider to accelerate more rapidly for a given value of h, thus giving the impression that gE was slightly greater in the laboratory. A local gravitational fluctuation can be ruled out, because though the value gE can vary depending where on the Earth’s surface a measurement is taken from, the difference is usually between 9.78 and 9.83ms-2 (i.e. not enough to cause a local gravitational acceleration of 10.73ms-2.

The inaccuracy in the time values could be remedied by adjusting the experimental procedure to avoid a non-zero initial velocity. If the photogates could be attached directly to the air track without disturbing airflow, then the issue to adjusting the position of the photogates after every increase in height would be eliminated. Also, the glider could be held at the start point by some mechanism that is also attached to the air track (so it too would not have to be adjusted after


every measurement), which can release the glider without applying any unwanted kinetic energy. Taking extra time to check that the surface where the experiment will take place is horizontal is essential. An uneven surface will, as previously discussed, will systematically return incorrect values for time t. As for changes in gE due to height above sea level, these can be ignored as the variation in gE with height will be negligible at any point on Earth’s surface.

5 Conclusion

An experiment has been conducted to ascertain the value of gravitational acceleration on Earth using a glider on a frictionless inclined air track. A value of 10.73±0.32ms-2 was calculated from the raw data using weighted linear regression and some error combination techniques. All of the data analysis was conducted using the computational program, SciLab.

A chi-squared test was also conducted to ascertain whether the calculated value supported the accepted value of gE (9.81ms-2). The test rejected the calculated value as supporting evidence for the value of gE due to an extremely high chi-squared value.

The main sources of error were attributed to a non-zero initial velocity (human error) and a possible increased incline of the air track due to an uneven workbench (systematic error). Possible solutions to these errors were also discussed.

Acknowledgements

The author would like to thank Miss. Rose Cooper and Mr. Andrew Johnston for their collaborative efforts in running the experiment and analysing the data. The author would also like to extend their gratitude to Dr. Balázs Pinter for his tutoring on regression analysis and error combination, which was paramount to the analysis of the experimental data.


References

1. Gravity – A definition

http://www.engineeringtoolbox.com/accelaration-gravity-d_340.html http://hyperphysics.phy-astr.gsu.edu/hbase/grav.html#grav

2. Value of gravitational acceleration on Earth –

CODATA Value: Standard acceleration of gravity. The NIST Reference on constants, units and uncertainties. US National Institute of Standard and Technology. June 2011, Retrieved 2011-06-23.

3. Newton’s Second Law –

John W. Jewett, Jr. and Raymond A. Serway (2010) “Physics for Scientists and Engineers with Modern Physics, 8th Edition” Cengage Leaning, pp.107-108. ISBN-13: 978-1-4390-4875-7.

4. Displacement equation (Simple 1D kinematics) –

John W. Jewett, Jr. and Raymond A. Serway (2010) “Physics for Scientists and Engineers with Modern Physics, 8th Edition” Cengage Leaning, pp. 42. ISBN-13: 978-1-4390-4875-7.

5. Regression analysis, error calculation and chi-squared testing –

Dr. Balázs Pinter (2012) “Regression Analysis/Chi-squared testing” and “Error Analysis” lecture slides, University of Wales, Aberystwyth.


A Raw and Linearized Data with Errors Table 1: Raw data readings from experiment

Height (m) (±0.002m)

Time taken (s) (±0.0001s) 1 2 3 4 5 Average

0.005 5.6740 5.6773 5.5976 5.6017 5.5674 5.6236 0.010 4.2361 4.2557 4.2830 4.2240 4.2151 4.2428 0.015 3.4225 3.4055 3.4658 3.4639 3.4423 3.4400 0.020 3.0114 2.9934 2.9981 3.0087 3.0188 3.0061 0.025 2.5746 2.5793 2.5832 2.5880 2.5860 2.5822 0.030 2.4810 2.4812 2.4478 2.4564 2.4583 2.4649 0.035 2.3349 2.2966 2.2783 2.2403 2.2364 2.2773 0.040 2.1254 2.1262 2.1066 2.1081 2.1217 2.1176 0.045 2.0199 2.0475 2.0458 2.0386 2.0455 2.0395 0.050 1.9358 1.9484 1.9533 1.9431 1.9470 1.9455 0.055 1.8487 1.8077 1.7997 1.8037 1.8025 1.8125 0.060 1.7409 1.7277 1.7227 1.7167 1.7154 1.7247 0.065 1.6936 1.6851 1.6859 1.6777 1.6903 1.6865 0.070 1.6355 1.6212 1.6163 1.6114 1.6142 1.6197 0.075 1.5615 1.5435 1.5351 1.5467 1.5392 1.5452 0.080 1.4959 1.4912 1.5063 1.5104 1.5048 1.5017

Table 2: Linearized data with unique errors Height (m) (±0.002m) y-value (s-2) Error on y (s-2) 0.005 0.0316207 ±0.0011140 0.010 0.0555514 ±0.0004816 0.015 0.0845051 ±0.0002587 0.020 0.1106606 ±0.0001739 0.025 0.1499755 ±0.0001114 0.030 0.1645892 ±0.0000972 0.035 0.1928233 ±0.0000772 0.040 0.2230037 ±0.0000626 0.045 0.2404100 ±0.0000562 0.050 0.2642029 ±0.0000490 0.055 0.3043995 ±0.0000400 0.060 0.3361808 ±0.0000348 0.065 0.3515825 ±0.0000326 0.070 0.3811806 ±0.0000291 0.075 0.4188231 ±0.0000255 0.080 0.4434387 ±0.0000235


B SciLab Coding for Data Analysis //Data analysis for lab report h=[.005 .010 .015 .020 .025 .030 .035 .040 .045 .050 .055 .060 .065 .070 .075 .080];//Height of air track t=[5.6236 4.2428 3.4400 3.0061 2.5822 2.4649 2.2773 2.1176 2.0395 1.9455 1.8125 1.7247 1.6865 1.6197 1.5452 1.5017];//Average time to 4 dp. //Plot of Height against time with error bars plot2d(h,t,style=-2) errbar(h,t,.01*ones(t),.01*ones(t))// Use 0.01 for error. //lineraize the equation and plot. y=t.^-2; x=h clf;plot2d(h,y,style=-2) // //Perform regression analysis on the data. coeffs=regress(h,y); function y=gravity(h) y=coeffs(2)*h+coeffs(1) endfunction // //Need to consider errors on data, they need to be combined before being added to the regression. erry=sqrt((((4./t.^6))*(0.1^2))+(((1./t.^4))*(0.005^2))) plot2d(h,gravity(h)) plot(errbar(h,y,erry',erry')) // //Weighted regression. function coefficients=weightregress(vX,vY,vSig) //where vX vY is the vector of x and y elements respectily and vSig is the vector of errors. s=(1./vSig.^2)'; S=sum(s); xbar=sum(vX.*s)/S ybar=sum(vY.*s)/S xybar=sum(vY.*vX.*s)/S x2bar=sum((vX.^2).*s)/S m=(xybar-(xbar*ybar))/(x2bar-(xbar^2)) c=ybar-(m*xbar) coefficients(1)=c; coefficients(2)=m endfunction // //Weighted error on data. function werr=weightederror(vX,vSig) s=(1./vSig.^2)'; S=sum(s) xbar=sum(vX.*s)/S x2bar=sum((vX.^2).*s)/S werr(1)=1/sqrt(S*(x2bar-(xbar^2))) werr(2)=sqrt(x2bar)*werr(1) endfunction //Enter required code into the console to produce figures. coeffs2=weightregress(x,y,erry') weightederror(x,erry') // //Chi squared test. function ypred=theoretical(Vx) ypred=4.905*Vx endfunction pred=theoretical(h) observ=y plot2d(h,theoretical(h),style=5) //Null hypothesis: The measured value is supportive of the accepted value (accept). //Alternative hypothesis: The measured value is not supportive of the accepted value (reject). chi2=sum(((observ-pred)/(erry'))^2) //434.99922 //Critical chi2 value=21.064 //Reject null hypothesis


C Look-up Table for Chi-Squared Test

Look up table taken from "Regression Analysis/Chi-Squared Test" lecture notes by Dr. Balázs Pinter

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Measurement of Earths Gravitational Acceleration … rearranged into the linear equation format ... 4 Discussion −2 … = (−) # $ %% & ' ((> -1, . ) ... Measurement of Earths Gravitational